The North River Passenger Terminal, or New York Passenger Ship Terminal, or Piers 88, 90 and 92, is in the 50's on the West Side.
It was falling apart even back in the 1980s, but is undergoing a significant renovation. Partly, to handle ships up to three times
the size it was designed for 70 years ago. Cunard and Carnival, and possibly others, have moved their operations to a new terminal in Brooklyn.