Mitch's answer provides an excellent summary of Barth's views on the issue of the English word "person," as well as a helpful quote from Barth's Christian Dogmatics. To supplement that, I'll reference theologians who have analyzed Barth's argument and dealt with this question.
Barth's foremost critic from the conservative side was Cornelius van Til. Van Til devoted two books1 and numerous articles to critical analysis of Barth's writings, and finds Barth's system to be "antitheistic":
His system of doctrine does not present to us an essentially Reformed or Christian viewpoint with divergencies here and there. His system of doctrine springs from an antitheistic root and presents some external similarities to the Reformed point of view but never on any point agrees with Reformed theology.2
Given such an understanding of Barth's work, it's not surprising that Van Til finds Barth's formulation of the trinity problematic. He objects to Barth's "modes of being" interpretation of the persons of the Godhead, and argues that Barth believes that the "centers of self-consciousness" orthodox position leads to tritheism.
Barth's rejection of the "centers of consciousness" understanding of the persons of the Godhead was also criticized by Leonard Hodgson and others holding to social trinitarianism.4
Some Reformed theologians did not see anything wrong with Barth's views of the Trinity. Louis Berkhof writes:
This view of Barth is not a species of Sabellianism, for he recognizes three persons in the Godhead. Moreover, he does not allow for any subordination.5
Similarly, Gregg Allison calls Barth's "modes of being" formulation "not the modalism of earlier heresy," and argues that Barth understood that "any doctrine that emphasizes the unity of the Godhead over the distinctions of modes or persons [...] is incorrect."6
Other evangelical theologians, like John Frame7 and Alister McGrath,8 mention the controversy regarding Barth's views on the Trinity in their systematic theologies, but do not take a stance.
Van Til has been criticized, I think legitimately, for his polemic approach to Barth, but given the centrality of the doctrine to Christianity and the extent of Barth's critique of fundamentalism, his concerns have merit. Still, most evangelicals seem likely to agree that Barth's divergence from orthodoxy on the question of the Trinity, if any, is minor compared to his views on Revelation and Christology.
Phillip R. Thome's Evangelicalism and Karl Barth is an interesting introduction to the conflict between Barth and his evangelical contemporaries, but he does not focus on the issue of the Trinity.
- The New Modernism and Christianity and Barthianism. For more, see Thome, Evangelicalism and Karl Barth, 35
- Review of The Karl Barth Theology, appearing in Christianity Today, 1931. (source)
- Van Til, "Has Karl Barth Become Orthodox?", Westminster Theological Journal, 1954, v 16, pages 135–181 (source)
- Garrett, James Leo, Systematic Theology, I-326
- Berkhof, Systematic Theology, 1.1.8
- Allison, Historical Theology, 249–50
- Frame, Systematic Theology, 476
- McGrath, Christian Theology, ch. 10