Assuming Ms Besant is correct that the burden of proof is on the one who affirms the existence of God, one might ask what sort of evidence she would accept as sufficient to convince her that God exists. The request for evidence sounds reasonable but is in fact often disingenuous. What the atheist often insists upon is not evidence but proof, and nothing short of a visual appearance by God would be sufficient to satisfy this demand. Indeed, Jesus Himself says that even were a man to rise from the dead skeptics would still not be persuaded (Luke 16:31).
Most people who disbelieve do not do so for want of evidence. Indeed, evidence is all around us: The information-rich biosphere, the astonishingly precise values of the basic parameters of the universe, the irreducible complexity of structures and processes found in living things, the extraordinary uniqueness of planet earth, the existence of human consciousness, the fact that the universe seems to have had a beginning, the origin of life from non-living matter, all of these point to an intelligent architect behind the cosmos we live in. Moreover, theism explains the human longing for meaning, morality, and justice better than does atheism and explains our existential experience better as well (e.g. our sense of guilt, freedom, self, and personal continuity are all more difficult to explain given a materialistic view of man).
Each of these may turn out to have a plausible non-theistic explanation, but until one is forthcoming, they must count as strong evidence that the universe is not simply an accident or the product of blind random processes. Richard Dawkins, the atheist biologist, is famous for having acknowledged the power of the temptation to ascribe the design of living things to an intelligent provenience. Unwilling to accept that conclusion, however, he wrote that scientists must constantly remind themselves that the design they see in living things is only apparent, not real.
If disbelief is not a matter of insufficient evidence then to what should it be attributed? For many, their disbelief is simply due to the fact that they find the possibility of a God in some way philosophically distressing and prefer not to accept a belief which makes them uncomfortable.
Numerous writers have noted, for example, that much of the impetus for the refusal to accept the Big Bang explanation for the origin of the universe until the evidence became so overwhelming that it could no longer be denied was due to the discomfort many materialist physicists experienced at the thought that the universe actually had a beginning rather than being eternal. A beginning suggests a cause, and a cause of something like a universe strongly suggests something akin to the God of traditional theism. For many this was philosophically and psychologically unacceptable.
The demand for evidence is a smoke screen often employed to mask the fact that the individual simply doesn't want there to be a God. As Pascal observed, there is enough light for those who only desire to see, and enough obscurity for those who have a contrary disposition (Pensees #430). That belief is a matter of one's disposition toward or away from the object of belief is apparent in that even materialists often believe many things in the absence of evidence or in cases where evidence isn't even possible.
For example, Annie Besant maintained that one shouldn't believe anything for which there is insufficient evidence, yet she believed that this maxim itself is true. Where is the evidence for it? Doubtless she believed that love is better than hate and kindness is better than cruelty. Did she have evidence beyond her own subjective preferences for these beliefs? Presumably she believed that the world has a past, a present and a future, and that the world is more than five minutes old, but it's hard to imagine what would count as "proof" for any of this. Perhaps she believed she and others have minds, but if so, how would she have convinced a skeptic? How would one prove a logical axiom like If A then not ~A? What evidence could one adduce in support of the claim that there is a world external to ourselves and that it's not just all in our minds? Perhaps each of these things can be demonstrated by a sufficiently clever philosopher, but the point is that one is justified in believing them even if one cannot think of any compelling evidence for the belief. We believe them essentially because it helps us to make sense out of our experience not because there is an abundance of objective evidence for them. Why, then, is belief in God different?
Perhaps it's different because in all the other cases we are either indifferent to, or anxious to accept, the belief in question, but in the case of God there is an antipathy toward the belief that impels us to erect what we think are insuperable barriers to it. As long as no one can prove God exists the atheist feels rationally justified in believing that he doesn't.
Yet, belief in God is not irrational. Many scientists believe that there are other universes besides our own, many believe that matter is comprised of infinitesimally tiny "strings," some believe there are living beings in the universe. They believe these things even though there's not a single substantial piece of evidence for any of them. They devote their lives to the study of string theory or astrobiology, the latter being a scientific discipline, a wit once observed, without any subject matter. Yet belief in the existence of a superintending mind behind the cosmos, for which there is enormous circumstantial evidence, is dismissed as superstition.
One sometimes wonders if the greatest sin, the unforgivable sin, is to just not want God to exist and to resist believing he does with all one's might.