As a tourist you are very unlikely to find yourself in the bad area's, but that's not to say there are some areas less touristy than others.

Places to really avoid are around Brownsville and East New York, but as a tourist I can't imagine you would find yourself out there in the first place.

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Manhattan is perfectly safe, however you might find yourself in the Southern edge of Harlem into East Harlem (top end of Central Park) where it is less desirable.

Generally anywhere in Manhattan from the top edge of Central Park down will be just fine (although Times Square is a bit of a cesspit)

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In Brooklyn, anything from Prospect Park to the West or North is tourist suitable (plenty other fine areas, but less likely you will find yourself out there) - do get down to Coney Island though.

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If you go to a Yankees game don't worry about it being in the Bronx, it's very busy and it will be subway there and back.