If it were up to me, I wouldn't drive to New York. I would drive to a town close enough to Manhattan, park the car in a park-and-ride short term parking facility, and then take the train (either Metro-North or the Long Island Railroad) into Manhattan. But that's just me.
As for whether it would be better to stay in Brooklyn, the Bronx or Queens, it would depend upon what you want to do while visiting the five boroughs (for one should not forget Staten Island). If most of the places you want to visit are in Manhattan, then it would make a lot of sense to be as close to Manhattan as possible. That would mean either staying in Long Island City or Astoria (both in Queens) or staying in a hotel in Brooklyn. If you also want to visit sites in "the outer boroughs" (i.e. Queens and the Bronx), it might be best to be in Brooklyn, which is pretty much equidistant from both Manhattan and the outer boroughs.
To save even more money, if you could, instead of staying in a hotel, you could always use AirBnB to find a place.
Give me an idea of the sites you want to visit, etc. and I could probably narrow down the choices even more for you, but on the face of it, Brooklyn seems to be your best bet.