I'd like to apologize, I think I misunderstood your intent. I thought you meant that superheroes should go and find criminals who hadn't yet committed crimes and putting them away.
Well, then I'd say they are already doing that. You think Bruce doesn't try and get the Joker every time he breaks out of Arkham before he commits any crimes? Superman is definitely keeping tabs on Luthor, always. They just don't mention it unless it happens to be relevant to a story. Which goes back to my earlier theory that it's just (supposedly) boring for a reader to read, "Checked Luthor. Eating dinner." every few pages in Actions Comics. I'd personally love that kind of layering and attention to detail, but most writers aren't going to bother, and those that are will only do it to build up some sort of story about Luthor. I think it's probable that Supes is doing it anyway.
We recently saw that, a little bit, in the JLA comic when Hawkgirl checked up on Cheshire in jail. Of course, the point was that Roy was visiting Chesh, but you can tell that the JLA is keeping tabs on known criminals, at least when possible.
But the kind of criminals in the DCU aren't stupid. They know the heroes are on the lookout for them, so they know to be pretty careful when breaking out. Normally they don't just spontaneously make a run for it - they have a plan, probably some sort of violence. The reason Bruce usually doesn't catch the Joker before someone dies is because the Joker breaks out and does something quickly. There's not always enough time to pore through the clues and find him. Plus you have to assume that the Joker is pretty good at avoiding capture when he wants to.
Like I said, I've got nothing against these kind of stories. I'd love for detective cases where heroes set up underground sting operations and track down escaped criminals using logic and methodology - but I don't think they will. And I think the times that heroes failed to do so in the past is mostly the writer's not really thinking through the story.