The best options I know of are Liberty Harbor (closest to midtown in terms of travel time), Croton Point park in Croton-on-Hudson (maybe an hour or slightly more to midtown), and Floyd Bennett Field in Brooklyn (part of the Gateway NRA, on Jamacia Bay, maybe an hour and half to midtown including a decent walk from the campground to the bus stop at the other side of the park).
I have stayed at Floyd Bennet Field. It is affordable, and the RV sites--really just chunks of a large concrete parking lot--are spacious. There are no hookups, but there is a dump station and showerhouse on site. Getting there is an adventure in driving through the city; from the northish direction, I suggest just taking Flatbrush Ave down from the expressway.
In general, probably Liberty Harbor is the best for seeing the city.