By what main criteria does the mathematical community judge whether an open problem is important?
An open problem is important if its resolution affects many, diverse areas (indeed, the farther flung the impact, the more significant the problem).
In the case of the Riemann hypothesis, there are many theorems that are conditional on it (see Consequences of Riemann hypothesis), though they are largely in the broader realm of number theory.
Other problems, like the P vs NP problem could have consequences beyond automata theory that include cryptography (and, by extension, our entire
e-commerce industry), computational biology, manufacturing and optimization, etc.