Karl Barth opposed infant baptism in the final fragment of the Church Dogmatics IV/4 but he also opposed re-baptism! Barth believed that infant baptism (or paedobaptism) was a disorder that the church needed to correct by delaying baptism until a person desired to be baptized and gave their consent to it when asked. Karl Barth admired the strength of the American baptists but he opposed re-baptism, which is so common in many American baptist churches today. Barth believed it was the church's task to correct baptism, not the task of individuals who were baptized in a disorderly manor. Barth believed that infant baptism was a disorderly baptism but it was not an invalid baptism that should be corrected by re-baptism or replaced by a second baptism. An individual in a church should never be encouraged or compelled to be baptized a second time, or multiple times, because even a disorderly baptism is a valid baptism that must be taken seriously.

Karl Barth was baptized as an infant and refused to be re-baptized as an adult. Barth said he was not asked to be baptized, and was unable to reply as an infant, nevertheless, Barth insisted his baptism did happen, and although it was performed in disorder it was very much a valid baptism. Karl Barth's baptism did occur, even though it was not performed in the way he wished, it did happen and should not be replaced or redone at a later time. 

In 1965 when Karl Barth was 79 years old, he gave a lecture refuting infant baptism. Afterwards, Geoffrey W. Bromiley said "a member of a community teaching believer's baptism . . . had expressed to Barth her pleasure at his repudiation of infant baptism and asked him why he did not have the courage to be baptized afresh." [1] Barth replied:

Dear Mrs. Gowalezyk,

That seventy-nine years ago I was baptized as an infant--like the vast majority of all Christians from the third or fourth century on--is one of the many disorders from which the church suffers. I have opposed this disorder. But I have never maintained that baptism administered in this disorder is not valid baptism. What was said to me back then, unfortunately without my being asked or able to reply, was said, and therefore I do not see why I should replace that baptism by another and second one. I regarded and still regard it as more correct and important to take my one baptism very seriously. Part of this requires calling on the church to remedy the disorder in the future. 

With friendly greetings,

Yours,

Karl Barth [2]

For full disclosure, I'm PostBarthian and presbyterian, so I affirm infant baptism (unlike Karl Barth). I highly recommend reading W. Travis McMaken's The Sign of the Gospel: Toward an Evangelical Doctrine of Infant Baptism after Karl Barth to understand how infant baptism fits within a PostBarthian schema. 

Sources:

1. Barth, Karl. Karl Barth Letters 1961-1968. Trans. Geoffrey William. Bromiley. Grand Rapids, Mich: Eerdmans, 1981. 189. Print.
2. Ibid.
3. Header baptism icon source: wikipedia https://en.wikipedia.org/wiki/File:Baptism_logo.jpg