New York City has many areas where drones are banned, due to many surrounding airports and the busy airspace with aircraft that this creates. Below is a map courtesy of Mapbox that shows where you can and cannot fly your drone in New York City. The red is all the areas drones cannot be flown. Some great places you could fly your drone in the areas that they are not banned (as far as I know) are Central Park, Marine Park and Battery Park.
Yes you can fly near airports... you just have to notify them of your flight plans, nothing more. Although I don't know how they handle it in NYC. Something tells me it would be a lot tougher as they can deny you for safety reasons. See the following quote from the FAA web site...I can't speak for any land restrictions or illegal airspace laws in NY but flying within 5 miles of an airport is not an automatic ban, just to be clear. A person can contact the airport tower for permission to fly within those areas.