If a solution was found to the Riemann Hypothesis, would it have any effect on the security of things such as RSA protection? Would it make cracking large numbers easier?
If by 'solution' you mean confirmation or counterexample, then no. One could just assume the result, produce an algorithm whose validity requires the Riemann hypothesis, and use it to break RSA codes. Merely knowing if the Riemann Hypothesis holds or not doesn't help you construct any factorization method (although it can tell you a theoretical bound on how well a certain algorithm can run). It is possible, however, that in the process of resolving RH, we improve our understanding of related questions/techniques and use our improved knowledge to create algorithms that do have the potential to crack RSA codes.

Most mathematicians don't want to see RH resolved just for a yes/no answer. It is more important that research into the problem produces new mathematics and deep insights -- the resolution of the problem is simply one goal to reach and a yardstick to measure our progress. The situation is similar to the development of algebraic number theory with the goal of understanding higher reciprocity laws. Along the way a whole new interesting subject opened up spawning many decades of interesting mathematics, and the original motivation no longer holds center stage.
