St. Luke's Roosevelt has a birthing center with 3 delivery rooms, your doc has to be affiliated with the birthing center AND you must take heir childbirth prep class in order to have the option of delivering at the birthing center. The delivery rooms have tubs that you can labor in, but when it comes time to deliver the baby they make you get out of the tubs. that is what I was told/saw on my tour there 2 years ago. Double check to see if the rules have changed.
As far as I have heard, there are no hospitals that have a birthing tub option - a real pity. Look into homebirth if you are comfortable with that option. Go to www.waterbirth.org to learn more about options.
Here is another good website:
http://mommypoppins.com/ny-kids/birth-place-alternatives-.... St. Vincent's is no longer an option, BTW.
If you are willing to go further away from NYC, there is a great birthing center in Rhinebeck (hudson valley) at Northern Dutchess Hospital that has a birthing tub.