We're throwing "Crime" into the genre description because there is some element of that in there, but really this is flat-out Drama. Calling it anything else feels dishonest. Like calling it a comedy because of the way Heywood pronounces "Dumas" or calling it a romance because of Andy's relationship with Bogs.
The story of an innocent man falsely accused and sent to prison for a crime he didn't commit is dramatic. The plight of inmates being treated inhumanely is dramatic. Andy's daring and triumphant escape is dramatic. Finally, we're sorry, but no matter what's happening on-screen, when you've got Morgan Freeman doing a voiceover, you are watching a drama. That guy could make us tear up during a car insurance commercial.