I used to live in the city and was always searching for the easy access wing-able launches. Turns out the Hudson River at 125th St. was the ticket for me being a 5 minute walk from my apt:
My advice is to approach the Hudson and the inner harbor waters with extreme caution. These waters are daunting b/c the current, ebb/flow of the tide, pollution, and large boat traffic introduce quite a bit of complexity into navigation. Make sure you have a backup plan if something goes wrong. It also helps to ensure the wind is strong and (for the Hudson) aligned SSW or NNE. Show the harbor police your NYC Parks Kayak/SUP permit.
Plumb Beach was my other default, as the car and the subway were 1-1.5hrs. Avoid low tide, otherwise you will be walking a long way out. There's often seaweed in the shallows off the beach to the SW. Most wind directions are okay. Bay chop and decent rollers form with stronger SE winds. Lots of local wind-surfers and, unless it's offshore, kiters. Hardcore crew shows up year-round.