score:7
Yes. In fact, one of Barth's lasting contributions was a recovery of the doctrine of the Trinity as central for theology, in reaction to German liberalism's denial or, at best, sidelining of the topic (Schleiermacher's The Christian Faith is a prime example of this—in an 800 page systematic theology, the Trinity only merits 14 pages tagged on at the end). Barth's Church Dogmatics begins with the Trinity and its entire structure is based around it.
Some have accused Barth of modalism because he used the term "modes of being" or "ways of being" (Seinsweise) to describe the different members of the Godhead, instead of the traditional "persons." This accusation is based on a misunderstanding, though an understandable one. Barth's refusal to use the term "person" is due to the isolated individualistic connotation that term has taken on in recent times—a connotation that he doesn't think was contained in "hypostasis" as used in the creeds. In Volume IV of Church Dogmatics he differentiates between these "ways of being" more thoroughly, making it clear that he is not, in fact, a modalist.
I'll quote here from CD I/1, §9.2 (page 360 in the Hendrickson edition):
What we have here are God's specific, different, and always very distinctive modes of being. This means that God's modes of being are not to be exchanged or confounded. In all three modes of being God is the one God both in Himself and in relation to the world and man. But this one God is God three times in different ways, so different that it is only in this threefold difference that He is God, so different that this difference, this being in these three modes of being, is absolutely essential to Him, so different, then, that this difference is irremovable. Nor can there be any possibility that one of the modes of being might just as well be the other, e.g., that the Father might just as well be the Son or the Son the Spirit, nor that two of them or all three might coalesce and dissolve into one. In this case the modes of being would not the essential to the divine being. Because the threeness is grounded in the one essence of the revealed God; because in denying the threeness in the unity of God we should be referring at once to another God than the God revealed in Holy Scripture—for this very reason this threeness must be regarded as irremovable and the distinctiveness of the three modes of being must be regarded as ineffaceable.
The modalist heresy held that God was only one (monarchianism), but that He revealed himself in three different ways. We might say that modalists hold that God is one "way of being" but three "ways of acting"; Barth, on the other hand, makes a clear distinction between the three "ways of being." Earlier on this same page, in fact, he argues explicitly against the idea that the Triune nature of God consists only in three different types of revelation.
The full argument in I/1 §9.2 is worth reading if you have the time and inclination.
Upvote:5
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.