(Updated on June 6th, 2016: Karl Barth's source letter
has been translated and added as an appendix.)
Karl Barth is infamous for his statements against homosexuality in his Church Dogmatics, Vol. III/4 (CD III/4) where he called it a "malady" and a "physical, psychological and social sickness" and a "phenomena of perversion, decadence and decay" and other colorful phrases. Near the midway point in the Church Dogmatics, there is a small print section of CD III/4 that contains these strong denunciations of homosexuality (quoted below) and is frequently cited to oppose homosexuality today. However, this is not Barth's final word on homosexuality (according to the renown Barth scholar George Hunsinger) because Barth changed his mind on this controversial subject (near the end of his life after abandoning the Church Dogmatics) due to his discussions with medical doctors who provided Barth with modern scientific research on homosexuality that Barth thought should be considered when interpreting the plain sense of the scriptures. First, I will review Barth's No and Yes to Homosexuality.
Karl Barth's infamous rejection of homosexuality in CD III/4
CD III/4 is the final volume in Karl Barth's Doctrine of Creation and is devoted to Ethics—in other words, it explains how Barth's Doctrine of Creation should be applied to the Christian life and it includes many fascinating ethical loci, such as his rejection of capital punishment, and discussion on self-defense, suicide, prayer and marriage. Barth's strongest statements against homosexuality are located here in a small-print paragraph in the middle of a problematic section of CD III/4: §54 Freedom in Fellowship 1. Man and Woman.
Students of Barth have said that Barth's views of women in CD III "needs to be corrected" and "his views of women in volume three are not essential to his project" (for example) and I believe this is justified based on what Barth said about women in this volume. Therefore, I believe that many people who cite this small-print section against homosexuality would at the same time reject Barth's view of Man and Woman that are the foundation of this small-print section. How then shall we cite this small-print section to dismiss homosexuals and yet criticize Barth's opposition to women's rights? At this point, I am unable to resist criticizing Barth as well, because I do not believe that the Church Dogmatics would dwell in the same ivory tower as the Summa Theologica if Charlotte Von Kirschbaum (CVK) had followed the ethical imperatives of CD III/4 and had remained in the proverbial kitchen as CD III/4 directs instead of contributing to the Church Dogmatics as she did. Barth invited CVK to participate in the Church Dogmatics in a time (late 1940's to early 1950's) that women were rarely allowed to do more than secretarial work in seminaries. So anyone who stands upon this small-print section against homosexuality is stumbling.
For full disclosure, here are Barth's colorful criticisms of homosexuality from the controversial small-print section referenced above:
These first steps may well be symptoms of the malady called homosexuality. This is the physical, psychological and social sickness, the phenomena of perversion, decadence and decay, which can emerge when man refuses to admit the validity of the divine command in the sense in which we are now considering it. In Rom. 1 Paul connected it with idolatry, with changing the truth of God into a lie, with the adoration of the creature instead of the Creator (Rom 1:25).
"For this cause God gave them up unto vile affections: for even their women did change the natural use into that which is against nature: and likewise also the man, leaving the natural use of the women, burned in their lust one towards another; men with men working that which is unseemly, and receiving in themselves the recompense of their error which was meet" (Rom 1:26-27).
From the refusal to recognize God there follows the failure to appreciate man, and thus humanity without the fellow-man (CD III/2, p 229ff). And since humanity as fellow-humanity is to be understood in its root as the togetherness of man and woman, as the root of this inhumanity there follows the ideal of a masculinity free from woman and a femininity free from man. And because nature or the Creator of nature will not be trifled with, because the despised fellow-man is still there, because the natural orientation on him is still in force, there follows the corrupt emotional and finally physical desire in which—in a sexual union which is not and cannot be genuine—man thinks that he must seek and can find in man, and woman in woman, a substitute for the despised partner. But there is no sense in reminding man of the command of God only when he is face-to-face with this ultimate consequence, or pointing to the fact of human disobedience only when this malady breaks out openly in these unnatural courses. Naturally the command of God is opposed to these courses. This is almost too obvious to need stating.
It is to be hoped that, in awareness of God's command as also of His forgiving grace, the doctor, the pastor trained in psycho-therapy, and the legislator and judge—for the protection of threaten youth—will put forth their best efforts. But the decisive word of Christian ethics must consist in a warning against entering upon the whole way of life which can only end in the tragedy of concrete homosexuality.
We know that in its early stages it may have an appearance of particular beauty and spirituality, and even be redolent of sanctity. Often it has not been the worst people who have discovered and to some extent practiced it as a sort of wonderful esoteric of personal life. Nor does this malady always manifest itself openly, or when it does so, it obvious or indictable forms. Fear of ultimate consequences can give as little protection in this case, and condemnation may be as feeble a deterrent as the thought of painful consequences in the case of fornication.
What is needed is that the recognition of the divine command should cut sharply across the attractive beginnings. The real perversion takes place, the original decadence and disintegration begins, where man will not see his partner of the opposite sex and therefore the primal form of fellow-man, refusing to hear his question and to make a responsible answer, but trying to be human in himself as sovereign man or woman, rejoicing in himself in self-satisfaction and self-sufficiency. The command of God is opposed to the wonderful esoteric of this beta solitudo [blessed solitude]. For in this supposed discovery of the genuinely human with woman, or as a woman with man. In proportion as he accepts this insight, homosexuality can have no place his life, whether in its more refined or cruder forms. 
A small criticism of this very small-print
What then are we to say about these things? (c.f. Rom 8:31) Why has Barth said "natural" so much here and now? Has Barth reverted to a Natural Revelation in this section due to his appeals to "nature or the Creator of nature" and humanity's "natural orientation"? In Barth's magnificent refutation of Natural Revelation in CD II/1, he boldly rejected the appeal to Romans 1:20's use of the word Θειότησ ("divine nature") because it was an hapax legomena (i.e. a word used only once in the bible). Why wasn't Barth as careful in this rejection of homosexuality based on Romans 1 as he was in his rejection of Natural Revelation based on Romans 1? Many have used the plain sense of scripture to assert that the theiotas of Romans 1:20 proved Natural Revelation. Barth is certainly not affirming a Natural Revelation, but what may we make of Barth's 'natural' arguments without jettisoning CD II/1 (which Hans Urs Von Balthasar considered to be the best volume in the entire Church Dogmatics!) I believe this criticism is not insurmountable (and may be addressed by a loyal Barthian reader) but is worth identifying.
In Barth's defense, his project in CD III/4 titled "fellowship in freedom" is on the right track, despite his missteps due to the time and place where he lived. I don't wish to refute Barth, but only refute those who assert that these comments definitively demonstrate that Barth was opposed to homosexuality.
Barth's change of mind according to George Hunsinger
George Hunsinger is a world renown Karl Barth scholar, and has written several articles on homosexuality such as There is a Third Way, Thinking Outside the Box: (Part 1/4), (Part 2/4), (Part 3/4), and (Part 4/4), as well as a helpful chapter on the ordination of homosexuals in his book The Eucharist and Ecumenism: Let Us Keep The Feast. Although this is a summary of Barth's Offene Briefe 1945-1968), this summary is essential to the future discussion of homosexuality in the PostBarthian context.
"Thielicke criticizes Karl Barth (and rightly so) for the position he took on homosexuality in Church Dogmatics. However, like many others, Thielicke was unaware that Barth later changed his mind. In light of conversations with medical doctors and psychologists, Barth came to regret that he had characterized homosexuals as lacking in the freedom for fellowship. In the end he, too, found it necessary to interpret the plain sense of Scripture in light of advances in modern knowledge. (Barth and Thielicke, by the way, both played a role in decriminalizing homosexuality in German society.) (Barth, Offene Briefe, 1945-1968, Zurich, 1984, pp. 542-43.)". 
UPDATE: June 6th, 2016
As an appendix, I have provided an English translation of Offene Briefe (1945-1968) to substantiate George Hunsinger's quotation. Based on my reading of Hunsinger, in general, he is a reliable witness to untranslated Barthianisms! Therefore, anyone who quotes CD III/4 to oppose homosexuality without mentioning Barth's letter in Offene Briefe (1945-1968) has put forth an irresponsible hermeneutic. Has such a person tried to understand Barth here and now, or are they gathering ammo for an a priori prejudice that which to oppose?
Appendix: Karl Barth's letter
Dr. David Congdon has graciously translated the source letter from Offene Briefe (1945-1968) and has given me permission to share his translation! The original letter was written by Eberhard Busch on behalf of aging Karl Barth who was 82years old at the time. Congdon has provided the German source and translation on his website The Fire and The Rose.
This letter confirms what George Hunsinger said in the quotation above regarding Barth's change of mind on homosexuality later in life. It would have been preferable to have a source that was directly written from Barth's own hand in a publish work. If this raises any doubts, remember that Paul's letter to the Romans was written in the same way by his amanuensis Tertius (c.f. Romans 16:22)
Dear Mr. Italiaander!
Professor Karl Barth took note of your letter on June 10 and is pleased that, in your planned anthology on the issue of homosexuals and their social status and recognition, you thought to give space to his voice.
In fact he has already once expressed himself on this issue (Kirchliche Dogmatik III/4, 1951, 184f.)—though in a sense that probably would not be appropriate and suitable for that section of your anthology. Lest you view the predominantly negative attitude toward homosexual relations in that passage in a false or exaggerated way, the following was briefly mentioned:
1) That one has to understand and appreciate what is expressed there—only incidentally—against the background of the whole context of that passage: a context in which Karl Barth interprets the command of God given to human beings as creatures and in their creatureliness under one of several aspects, namely under the “freedom for community.” For him the original form of interpersonal community (not merely “marital” but all natural community) is the counterpart of man and woman.
2) In this context homosexuality in its essence appeared to him as a form of unfreecommunity—namely, as a behavior in which one closes oneself to and withdraws from one’s freedom for community. But you can be sure that his opinion on this point did not and does not imply as such a license for “defamation,” let alone for the (nonsensical) legal “punishment” of homosexuals (at least insofar as they do not “seduce” or “harass” others). For he does not consider them actually wicked but rather he considers it emotional Pharisaism when, on the one hand, there is a degrading of the articles of the law (though often not carried out to the same degree), but on the other hand in contemptuous whispers people take actions against them or create a hostile environment. By no means!
3) With respect to his former incidental remarks—in view of the changes and new discoveries that have occurred since its writing—Professor Barth is today no longer entirely satisfied and would certainly today write them somewhat differently. One may think, therefore, precisely against the background of the context in which God’s command fundamentally wants to be perceived and followed as “freedom for community,” that—in conversation with doctors and psychologists—one could come to a new evaluation and presentation of the phenomenon.
You would naturally now like to hear this from him. But having endured eighty-two years of all kinds of limitations, he now no longer has the time required for this purpose. They say that he should make use of his remaining strength to work on those themes and tasks that presently appear more important to him. We ask for your kind understanding!
Greetings on his behalf,
German text in: Offene Briefe 1945–1968 (Gesamtausgabe 5.15), 542–43. 
[^Header Image] Sappho and Erinna in a Garden at Mytilene 1864 Simeon Solomon 1840-1905 Purchased 1980 http://www.tate.org.uk/art/work/T03063 (Edited)
[^1] Barth, Karl. Ed. G. W. Bromiley and T. F. Torrance. Church Dogmatics III.4: The Doctrine of Creation. Vol. 19. London: T & T Clark, 2010. 159-60. Print. Study Edition. (paragraph and formatting were added for readability)
[^2] Hunsinger, George. "Thinking Outside the Box, Part 4: The Voice of ‘Progressive Traditionalists’." The Presbyterian Outlook. N.p., 13 Mar. 2002. Web. 05 June 2016. <https://pres-outlook.org/2002/03/thinking-outside-the-box-part-4-the-voice-of-progressive-traditionalists/>.
[^3] Congdon, David. "The Fire and the Rose." : Eberhard Busch to Rolf Italiaander, 1968. David Congdon, 06 June 2016. Web. 06 June 2016. <https://fireandrose.blogspot.com/2016/06/eberhard-busch-to-rolf-italiaander-1968.html>.