Take the crude evidentialist's axiom:
E: One should not believe that some proposition P is true without sufficient evidence.
Let's now assume that all of the arguments of natural theology are unsound. Let us further stipulate that there cannot be any sound argument of natural theology. Under such assumptions, there is no evidence for God's existence and there cannot be any such evidence. (Of course, I think these assumptions are mistaken.)
Taking these contentions at face-value, it follows that one should not believe that the proposition, "God exists," is true. However, a major difficulty arises when we begin examining some of our relatively uncontroversial beliefs. I'm thinking in particular of the following beliefs:
1. There exists an external world.
2. Minds other than my own exist.
3. The past has existed for more than five minutes.
The list can be extended much further, but it should be clear by now that none of these beliefs can be supported by way of evidence. How, for example, would one go about providing evidence that the past is more than five minutes, and that one's memories of an older past is not just illusory?
The problem is compounded further when we ask: what evidence is there in support of the E? If there is none, then E should be rejected on its own terms.
It seems, then, that the acceptance of these additional beliefs, if rationally believed, can be used as a part of a reductio ad absurdum against E. These rationally-held beliefs that happen to be non-evidence-based are called "properly basic beliefs." One is justified in believing that an external world exists, etc., even though he/she cannot prove via evidence or otherwise that the external world is not just illusory.
Plantinga and other philosophers have postulated that belief in God is also properly basic, and that belief in God can be rationally held with or without corresponding evidence. One obvious objection to this is that belief in God is not indispensable. A person may function just fine in society without believing in God. However, I think this objection falls short of being persuasive. I say this because other properly basic beliefs do not appear to be indispensable, either. If I were to adopt solipsism and believe that I am the only mind that exists, probably very little would change in my behavior. There are certain advantages in being kind, etc., even if altruism is illusory and it is for my own benefit alone. I would prefer to have pleasant illusions rather than unpleasant ones, after all.
Therefore, it seems quite unmistakable that indispensability is not the sole criterion for what constitutes a properly basic belief. As for belief in God being properly basic, there is likely something to that claim. Anthropology has shown an almost universal acceptance of God-belief among different cultures. This may suggest that theism is a naturally-held belief (a belief independent of specific culture), and if naturally-held beliefs are properly basic (are they?), it would follow that belief in God is properly basic.