Penn Station is the closest Amtrak station to JFK. If you wish, you can book the entire trip from JFK to Durham through Amtrak, as Amtrak offers connecting bus service from JFK to Penn Station. At www.amtrak.com, enter "JFK" as your origin, and "DHM" as your destination to see times and fares.
There are other ways to get from JFK to Penn Station. Check out the options at www.flyerguide.com.
A cheaper way of getting to Durham would be to take Bolt Bus or Megabus from Penn Station to Boston, and Amtrak from Boston's North Station to DHM.
Penn Station is the closest