I would say that 6 hours is too tight for a visit to the statue for several reasons. First of all JFK and the statue are exactly at the other ends of manhattan so you will have to croos all town and bridges/tunnels if you would consider by car (which i would not recommend you) the other reason is that you need to take a ferry to the statue island and that takes time and they have a program. Third reason is that your 6 hours is probably not net time as you will have to check in again way before yo9ur flight. Security and customs take time in NY. I was in May on JFK and experienced myself.
The best way to go downtown Manhatten in my opinion is to take the subway from JFK.(if i remember well it is the blue line...but they can advise you). Disadvantage is that it crosses the Bronx and Queens and i didnt feel really safe and comfortable but it brings you very fast to the herat of manhattan.
Good luck!
NY statue of liberty