Because string theory contains General Relativity in the low energy limit, it is possible to study black holes in string theory. Off course, to make this a rigorous statement, we would need a quantum theory of gravity.
String theory claims to be just that.
As a first step we should try to find classical black hole solutions to the string equations of motion. Because quantum effects are only expected to influence the classical solutions near the curvature singularity these classical solutions should be good approximations of the real solution away from the curvature singularity. In this light, the extremal solutions are particulary interesting. It is known that in General Relativity, the charged solution in the extremal state is stable against Hawking radiation. It is interesting to see what String theory has to say about that.