From the little of Karl Barth that I've read, here is a crash course on Barth and universalsim - the belief that every human being will be saved.
Nein! Why? Because a grace that must save all people could not be free grace; it could not be the grace of a God who is free.
And yet, as Barth asks, "would it be God's free grace if we could absolutely deny that it could do that?" It would not, and so to reject the possibility that every human being will be saved is to reject the nature of God's free grace; indeed, it is to reject the freedom of God to be and do what He wills. Barth says unequivocally that "there is no theological justification for setting any limits on our side to the friendliness of God towards humanity which appeard in Jesus Christ".
Barth has a timely word for those of us who are so antagonistic towards universalism. Listen to him:
Strange Christianity, whose most pressing anxiety seems to be that God's grace might prove to be all too free on this side, that hell, instead of being populated with so many people, might some day prove to be empty!