It is useful to the currency itself.
Bitcoin tries to be a democratic currency rather than being controlled by a central authority. Democracy means that the majority of people rule by their vote. This is a concept that might work in real life (all irony intended), but it's hard to pull off on the Internet since we can't simply prevent one real person from creating thousands of virtual identities and using them to make his vote louder (so called Sybil attack).
Bitcoin solves this via a concept known as proof-of-work, which requires some non-trivial work to be done for every vote to be registered. While it is not a perfect solution (surely one can manage to do twice as much work as somebody less capable), it has an ambition to prevent all serious scams, making it infeasible for a single entity to overpower (outvote) the crowd.
Mining is the process of voting, and miners are the voters. Every voter has to find a reverse hash according to some very specific rules, which is considered a hard work. All the others can verify his solution easily.
I know this does not describe the problem entirely (and it was not my goal), but it should give you the idea. If interested in the technical details, please see the original paper or read some articles explaining Bitcoin in-depth. There's plenty of them out there, but be prepared to spend a few hours going through the stuff.