If I remember correctly, he does help Gotham with his money by donating to schools and making building projects. But I think even with this way, it won't be able to save Gotham. Crime will always be there no matter what.
Always criminals in all classes who want the world to burn it is a moral problem more than just financial one. Like you say why are their rich criiminals and revolutionaries. The desire for power
My take is that he can't or at the very least can't make huge changes to Gotham since it would make him a target for crime lords and secret societies.
In Batman Begins, if I remember correctly, his father was killed for trying to save Gotham and I think they mentioned his grandfather was also killed when he tried to help Gotham.
In Batman Earth 1, Penguin got his parents killed because he knew that Thomas would rid crime in the city.
The Court of Owls owns Gotham so any attempts would be sabotaged.
So he needs to first try to get rid of all that before actually making any real systemic changes to Gotham.
He does do this. His uses his money to fund being Batman.