Liberty Harbor is probably the best bet all around.
I recently (over Memorial Day weekend) stayed at Floyd Bennet Field in Brooklyn, part of Gateway NRA, and had a good time of it. Getting there does involve driving through at least some of the New York street system; from the north, I would suggest I-95 to I-278 to Flatbrush Ave all the way down as doable if not enjoyable. Obviously rush hours are best avoided.
RV camping at Floyd Bennet Field is also more or less a parking lot, with much bigger spaces than Liberty Harbor and no hookups (though there is a dump station). There's a bus to the subway at the entrance to the park as a whole, which is a decent walk or quick drive from the camping area. It's maybe an hour and a half on average to get to midtown Manhattan, depending on how the subway, train, and your schedules all mesh together.