This is a far simpler question than it first seems, but not for the reason usually assumed! Regardless of one's beliefs or experiences of God, it can be interesting to explore this issue as a matter of philosophy of language.
The answer to this question obviously depends upon what one means by "God" and "exist". When we ask "Do unicorns exist?" we are asking if there is a physical animal corresponding to the idea of the unicorn. But if we ask this same question about God, then clearly God does not exist since no-one claims that God is a physical entity (except, oddly, Dawkins in his complexity disproof).
So what does it mean to say that a non-physical entity exists? Does Switzerland exist? Well, yes and no; there is a physical landscape we call Switzerland, but the country itself is a concept of mind, a mental model of its borders and people (what Benedict Anderson calls an imagined community). Does Einstein's theory of general relativity exist? No, it's a concept of mind that models physical events. Similarly, for most people who believe in God, the word "God" is a concept of mind that models spiritual events. Paul Tillich sees God as "the ground of being", John McQuarrey as "being itself" and Emmanuel Levinas as "the unknown and absolute other". Not one of these ideas of God implies any claim to existence in the way this term is usually used.
So no, God does not exist. But this statement has nothing much to say about God.