Neither. It's more complicated than that. 
Gotham and Metropolis are different cities with different problems. There is not consistent enough street crime to merit Batman's presence in Metropolis. It's a hub for weird science and global conspiracies led by white collar criminals and research teams that lack proper ethics. Gotham's issues are more rooted in systematic corruption within the community itself that manifests as an absurdly active criminal underworld. Superman could THEORETICALLY clean up Gotham's streets, but it would take him all night, and the same thing would happen again the very next day until somebody deals with the root of the problem. They are both the right kind of hero for their natural environment.