Basically, you have three choices:
1) As IAAL said, you can educate yourself on the legal process of judgment enforcement. One way to do this is research at a law library or you can go to any large bookstore and look for a book titled something like, "How to win in New Yord courts". The book will include a chapter on how to collect.
2) Get hold of a Judgment Recovery or Enforcement firm. You could do a websearch for firms in New York (or go to my website shown in my address line and I will try to refer you to someone based in New York).
3) Let the debtor continue not paying.