As someone who has yet to visit New York City (still near the top of my bucket list), I in no way feel I can say what director in the history of cinema has best captured the city.
A few names do come to mind: Martin Scorcese, Nora Ephron, Sidney Lumet, and Woody Allen (despite his troubled personal life). Then of course there is Spike Lee. While he has not entirely relied on New York for all of his films, his best ones (namely Do the Right Thing in 1989) have helped shape parts of the city that never sleeps in our movie going psyche.
Nothing else to say 