Why doesn't the Brooklyn Bridge have a train on it anymore?

No. The subway trains to and from Brooklyn run on their own elevated lines. They do not use the Brooklyn Bridge. The bridge is for cars, buses, trucks, and pedestrian traffic only.

There used to be trolleys that ran across the Brooklyn Bridge, but that was only from 1898 to 1950. The old BRT line also ran over the bridge and just barely into Manhattan, to its last stop, but that was around a hundred years ago.