Buzz Feed dug up a 1990 edition of "The Atlas Of The DC Universe" and it shows that Gotham City is on the north side of the Delaware River. This area would be part of New Jersey. However, even the DC Comics Wiki entry that Buzz Feed references says that various Batman writers and editors over the years have claimed that the famed Dark Knight's hometown of Gotham could be pegged geographically elsewhere.
In fact, according to Les Daniels, author of the 1999 book Batman -The Complete History, Gardner Fox, one of the original Batman comic book script writers, along with Bill Finger, for Batman creator Bob Kane, "had called Batman's previously unnamed town 'New York,' but Finger didn't like that, and struggled to come up with a substitute that had the right mythic ring. He considered Civic City, Capital City, and Coast City, before finally settling on Gotham City. 'We didn't want to call it New York because we wanted anybody in any city to identify with it,' he said."