Other than Liberty Harbor you are going to be a considerable distance from Manhattan and Brooklyn and you will need to travel strictly by public transportation for one or more hours with the commuters depending on time of day to get to Manhattan or Brooklyn. There are two campgrounds in Nassau County on Long Island that are run by the County. One at a beach and another in a wooded setting. The Lido Beach campground is about 45 minutes by train to Manhattan with a switch to the subway to get to Brooklyn. Battle Row Campground (no battleground was ever there) in Bethpage is about an hour from Manhattan by train with the same switch to the subway to get to Brooklyn.
Once you get to the campground just leave your RV there and do not even think about taking it into Manhattan or Brooklyn. These are not places to drive an RV or expect to park it - anywhere. Have the friends or relatives come meet you at the campground with their car and take you to where you all want to go. Long Island is not as bad with driving an RV (stay off the parkways due to extreme low clearances) but parking a large RV is also going to be a challenge to find a large enough parking lot other than malls and large shopping centers.
Yes, Liberty Harbor is a large parking lot but it is extremely convenient to Manhattan and Brooklyn and should be able to accommodate your RV. No matter where you go, campgrounds charge extra for visitors.