How do we resolve the issue with the circular argument that the Bible is the basis for its own authority?

It is circular in a sense, but all philosophical starting points are. I would contend that authority begets authority, and our philosophical foundations appeal to an authoritative source of truth and knowledge, whether we realize it or not.

In order to be the authoritative source of all things true, however, something either needs to authoritatively declare itself as the authority of all things true (If all things known are known via X; then that knowledge must also come from X…but this is circular) or a higher authority must identify it as the source of all things true. The latter scenario, though, simply moves the bump in the rug (as it were), because then the onus shifts to that higher authority to then go through the same motions that the previous one did (i.e. either {circularly} establish itself as the authority of all things true, or establish its credentials…by ultimately appealing to a {circularly} affirmed higher authority). Ultimately, we do have to start with some sort of tautological starting point, but the key is to be willing to honestly evaluate the consistency of our starting points and judge the aseity of our ultimate authority (can that authority stand on its own?).

To take a simple, and all too common (pun not intended), example, many people -- whether they fully realize it or not -- ultimately appeal to their understanding of "common sense" as their authoritative source of knowledge and reality. The problem is that -- aside from being a very subjective starting point -- "Common sense" doesn't definitively identify itself as the starting point of all things knowable and known. In fact, I would say "common sense" actually says that "common sense" is limited in its grasp of objective reality because it's so intrinsically dependent on personal interpretation and sensory stimuli. If common sense tells us there's a reality beyond that which we perceive and understand intuitively, on the one hand, then it's absolutely inconsistent to then appeal to common sense as our ultimate source of understanding.

So, while someone's free to appeal to their "Common sense" superficially, they're left lacking when it comes to establishing the basis for doing so in deeper philosophical contexts, because "Common Sense" is not a clear authority (even by its own right) of reality and there is no clear higher authority that grants it such authority by proxy.

A better (though, perhaps more controversial) example might be "science" Empirical science is a wonderful means of learning about the world around us, and it's an important discipline to nurture in order to refine our understanding of the physical realm, but some people appeal to it as final authority. The problem is that science itself doesn't stake such a claim, so if someone comes to that conclusion, it's based on something contra-science, which makes that philosophy immediately inconsistent (since it supposedly relies ultimately on scientific revelation).

Another example might be the "Flying Spaghetti Monster," Which as I understand it is a pretty good attempt at a reductio ad absurdum against the whole appeal to dogma, but the problem is that (assuming, for a moment, it's not meant tongue in cheek) the FSM never establishes himself as a final authority (AFAIK), so in order to appeal to him in good faith, one must still appeal to a higher authority which then inherits the onus of establishing itself as authority....otherwise, as a joke, it's kind of cute, but as a legitimate counter-offer, it doesn't really solve the problem it's so intent on identifying. (it's sometimes easier to mock people providing answers, it seems than to offer substantial answers on one's own)

That being said, the Christian Bible is never going to be irrefutably proven as a final authority, because, I think, it's philosophically impossible to do so, but I think it does satisfy this in some significant ways: It appeals to itself as the word of God who is identified as a God of revelation who does not lie.

Philosophically, it establishes that God is the source of creation who then grants man a portion of rationality (by creating him in His image). He then communicates to man through His word (revelation). These pieces together provide an objective framework against which we can justify deductive reasoning (which requires communicable true knowledge and a means of applying that knowledge to reach sound conclusions -- e.g. it presupposes a rational mind, and it presupposes rationality as a transcendent quality) as well as a basis for a healthy understanding of inductive empiricism (which presupposes a transcendent rule of order but acknowledges that our observations are influenced by imperfect sensory interpretations). It also accounts for an objective and qualitatively measurable morality (by contending that morality not "owned" by societies, but rather by the Author of all things, we actually are in a position to judge certain societal standards as lacking), it establishes consequences for man's transgression from the law of the Lawgiver, and provides the means of reconciliation, and does so on God's (not man’s) terms (e.g. the wages of sin is death; all have sinned; Jesus -- who knew no sin -- died for the sins of others that they may live)

I'm not sure that it's provable beyond that, and all of those premises require a bit of faith to accept. BUT, I think it does provide a very solid, objective foundation (and I would say uniquely so) for a consistent philosophical/moral framework. When the philosophical-reductionist conclusion is pointed out that it’s ultimately circular, I think we can accept that and challenge other frameworks to the same standard to see if they don't, indeed, have the same philosophical limitations (and I would say even more so).

tl;dr version:

The Bible asserts itself as God's word and also establishes God as authoritative, which does seem to beg the question a bit. To the Christian, It reveals His role as Creator and Redeemer and defines Him as the source of all things knowable and known; it reveals His plan for man and gives man an objective standard by which to orient his life. We all have to start somewhere philosophically, and as Christians, we think that this is a pretty good place to start. When the question is raised, I would challenge someone to offer a contrary philosophy that provides a more comprehensive “solution” without falling into the same "traps" they see of circular presuppositions… I don’t think it can be done.