I would pay decent money to see a debate between a resurrected Hitchens and Barth. Maybe some day, and maybe I won't even have to pay.
Popular atheism rests on a negative answer to the question, Does God exist? If I am reading Barth rightly - and that is always a large "if" -- then Christians cannot answer that question in the way that atheism (and Christianity) thinks that they must. We do not know what existence or being is, and then try to work our way to a god who ticks this box. God does not exist in the way that a human exists, or in a way that a fairy or a leprechaun or a piece of bread exists. God does not just have being, but has the very power of being. He reveals Himself to us as the One in whom we live and move and have our being, and because He reveals Himself to us as this One, we know that He is and that we are -- not in the same way, but in a quite distinct way. For us to try and move from "we are" to "He is" without the grace of revelation is either a hopeless exercise or an exercise in idolatry
For Christians, we only inquire into the possibility of God's existence on the basis of God's making Himself known through His Word, and on the basis that the One we inquire about is not a sharer in the thing we know as our existence, but is its Redeemer, Creator an Lord.