Is Karl Barth a universalist?
This is one of the most common questions posed to Barth, both during his lifetime and among his students to this day. For the sake of clarity, I am using “universalist” to refer to an ultimate restoration of all people. Barth is assuredly a “universalist” if we are speaking of the scope of the atonement or, as Barth would prefer to say, man’s “justification, sanctification, and calling” in Jesus Christ. However, the question of Barth’s “universalism” is more commonly in reference to whether he affirmed that all people, by the Holy Spirit, would receive our “proper being” in Jesus Christ. That is the sense in which we ask the question, “Is Karl Barth a universalist?”
Let us first take a look at an instance where Barth comes closest to affirming a universal restoration. In Church Dogmatics IV.1, Barth gives a survey of his christology in the doctrine of reconciliation. Therein we find his famous threefold delineation of Christ’s person and work: the servant as Lord (justification), the Lord as servant (sanctification), and the true Witness (calling). After mapping the territory which will occupy the rest of CD IV, Barth addresses the “subjective apprehension and acceptance” as distinct from the “objective relevance” of man’s justification, sanctification, and calling in Jesus Christ (147). He also uses the language of “appropriation” as distinct from “ascription,” the former of which is “the being and work of His Holy Spirit.” Barth recognizes that the “ascription” is universal but the “appropriation” is given according to the Spirit’s determination, and both are equally the work and decision of God: “That God did not owe His Son, and in that Son Himself, to the world, is revealed by the fact that He gives His Spirit to whom He will” (148).
As if that were not clear enough, he repeats, “In this special sense Christians and only Christians are converted to Him. This is without any merit or co-operation on their part, just as the reconciliation of the whole world in Jesus Christ is without its merit or co-operation” (148). Barth is comfortably Calvinist here. He refuses to introduce even the mildest synergism at the point of “appropriation,” just as surely as he refuses to do at the point of “ascription.” Yet, Barth will speak of Christians as “representatives” of all people, and this is where the universalism comes to the fore, implicitly at least:
[Christians] have over the rest of the world the one inestimable advantage that God the Reconciler and the event of reconciliation can be to them a matter of recognition and confession, until the day when He and it will be the subject of His revelation to all eyes and ears and hearts, and therefore of the recognition and confession of all men. 
The language here is the same that he uses for those who have been awakened by the Holy Spirit: “eyes and ears and hearts.” Barth had similarly drawn together the “elect” and “rejected” (Jacob/Esau, David/Saul, etc.) in CD II.2 with the use of “proximity” language, in order to emphasize their common orientation to Jesus Christ. Yet, I do not recall Barth coming this close to a universal restoration in II.2. In fact, he is at pains to avoid it when he discusses “the determination of the rejected” and in his massive excursus on Judas at the end of the volume.
When we turn later in the Church Dogmatics to IV.3.2, we have an instance of where Barth directly addresses the question of universalism and expressly rejects it, even though he recognizes that “theological consistency” may urge us in that direction. Here is how he discusses the matter:
A final word is demanded concerning the threat under which the perverted human situation stands, in spite of its limitation by the powerful and superior reality of God and man, to the extent that from below it is also continually determined by the falsehood of man in a sinister but very palpable manner. Can we count upon it or not that this threat will not finally be executed, that the sick man and even the sick Christian will not die and be lost rather than be raised and delivered from the dead and live? …
First, if this is not the case, it can only be a matter of the unexpected work of grace and its revelation on which we cannot count but for which we can only hope as an undeserved and inconceivable overflowing of the significance, operation and outreach of the reality of God and man in Jesus Christ. To the man who persistently tries to change the truth into untruth, God does not owe eternal patience and therefore deliverance any more than He does those provisional manifestations. We should be denying or disarming that evil attempt and our own participation in it if, in relation to ourselves or others or all men, we were to permit ourselves to postulate a withdrawal of that threat and in this sense to expect or maintain an apokatastasis or universal reconciliation as the goal and end of all things. No such postulate can be made even though we appeal to the cross and resurrection of Jesus Christ. Even though theological consistency might seem to lead our thoughts and utterances most clearly in this direction, we must not arrogate to ourselves that which can be given and received only as a free gift. [IV.3.2, p. 477, underlining mine]
Barth will immediately, on the following page, say that “there is no good reason” that we should not be open to the “unexpected withdrawal of that final threat” — “open to this possibility” (478). Yet, he has made it clear that it is not known, which is why he uses the language of “possibility,” whereas he loves to use “certainty” for the objective work of the God-man. As far as I know, this is the most explicit rejection of universal restoration in Barth’s Church Dogmatics. Whether this is entirely compatible with his statements in CD IV.1 is another question.