New York City

Being that you are from AZ (according to your profile), do plan your route in NY carefully. You are not permitted to be on anything labeled "parkway" in NY. It's common knowledge to those in the northeast, but definitely not to those from the southern or western states (case in point our neighbors in Texas and a truck pulling an Airstream I saw, both routed onto the Taconic Parkway which is NOT and RV-permitted route for a reason - designed for passenger cars only - no shoulder and clearances in many areas less than 9 ft.

Esyers is quite correct regarding Brooklyn vs. the Bronx and Floyd Bennett Field. I should spend a bit more time and energy actually thinking when posting, it seems. :S

I think DrewE meant to say Brooklyn and not Bronx in referring to Floyd Bennett. As he said, getting there can be challanging.

Liberty Harbor isn't fancy but it's right on the water with a great view of the skyline. It's also a short walk to transportation into the city and the various tours. The further out campgrounds could be an hour commute each way.

Liberty Harbor stays pretty busy so give them a call if that location interests you.

Happy and Safe Travels

Three options come to my mind, all more or less doable, but with quite different advantages and disadvantages.

Liberty Harbor, as mentioned, is for many people the most convenient and probably best option. It's not a fancy campground (think parking lot with utilities), but it is at a very handy location with easy access to mass transit. It's just across the water in New Jersey.

Croton Point county park (Westchester county, north of the city) is very close to a commuter rail station and thus fairly convenient to the city.

Finally, I've stayed at Floyd Bennett Field, part of the Gateway National Recreation Area, in the Bronx on Jamacia bay. It's about a mile walk (or drive) from the campground across the park to a bus stop, and thence to the subway, maybe an hour to an hour and a half all told to central Manhattan. There are no hookups, though there is a dump station and potable water and shower house available. The RV "sites" are just (large) marked-off areas at the perimiter of a big concrete expanse, a former seaplane base/tie-down area. Getting there is something of an adventure as the obvious route is via parkways that RVs are not allowed on; the best route, from the north at least, is probably to take the expressways to Flatbush Avenue clear across the Bronx.

You do not want to drive (and park) in New York City in lieu of taking tour busses or the subway. To see all the sites, plan on staying for a long time.

For visiting the city everyone seems to think Liberty Harbor is the closest and most convenient for accessing the city.
Personally I would stay an hour or more outside the city, and use the Towed to commute in or to access the trains.