I belong to a sailing club (Hudson River Community Sailing) that's at pier 66 (just a bit south of the Boat Basin). We sail (and race) J/24s all the time in the Hudson. Here's a few things:
1) There is a lot of commercial traffic. The tugs and barges, circle line boats, dinner cruisers, and cruise ships are the ones to watch out for. They are the stand-on vessel. The little water taxis like you to maintain your course and speed, and they'll go around you.
2) The tides can go pretty fast; over 3 knots. You should have an engine and a copy of the Eldridge Tide and Pilot Book (or just looks stuff up on NYHOPS: Urban Ocean Observatory at Davidson Laboratory
3) They're on the charts, but there are a few exclusion zones around the statue of liberty and the cruise ship piers by Manhattan north of the Intrepid.
4) The New Jersey side has some (soft) shallow areas. Don't go too close to shore. Also, don't go between the piers anywhere. You will likely hit a submerged piling.
5) The wind is mad shifty, and there's a dead zone near downtown Manhattan.
Regarding the boat basin, I think there's a stupid long wait (like over 5 years) to get a slip. Not sure about moorings. It goes get pretty rolly.
I love sailing on the Hudson, mainly because I live and work in NYC so it's super-convenient. But, it's rarely boring, and the views of the city are great!