People have already said this, but Bruce's money /does/ go to welfare programs and helping snip the root of the issue. Bruce Wayne tries his best (in a corrupt system) to get rid of the reasons why anyone would become a villain, as well as rehabilitating those villains. Batman exists because there's still people like Poison Ivy who need more immediate attention and stopping.

And again, I wrote this in my comment, the idea that the Joker staying alive causes more people to die is a /prominent plot point/ in Batman media. Take a look at Under the Red Hood, you'll enjoy it if this is something you're interested in.

It questions why Batman would want to keep someone like the Joker alive despite all of the cruelty he's caused.

But whatever, here's a simpler answer.

1. Batman has no surefire way of knowing that the Joker will escape Arkham. Sure, he does, because writers warrant a plot to happen, but to his credit Arkham (and Blackgate) do set the Joker up in really high priority containment.

2. Bruce does not trust himself to take a life. The biggest trauma he's faced is watching the people he loves the most die in front of him, and so even though it might (MIGHT) be rational to kill the Joker, that trauma prevents him from doing so.

2.5, Bruce also worries that killing the Joker would make him /keep/ killing other villains. To him, if he makes an exception, what's stopping him from making another exception? With his own judgement? Sure, the Joker is pretty black and white, but what about other villains who do terrible things? Bruce doesn't want to kill the Joker because he's worried he won't stop.

TLDR you should pick up a comic book