Cab fare from JFK would be expensive as you'd be on the meter. Best bet is by train to Little Falls (no direct service to Fairfield) and take a cab from there. To get to Penn Station take Airtrain from your terminal to Jamaica Station and either E subway or LIRR (faster but more expensive than subway) to Penn Station and take NJ Transit train. Little Falls is on the Boonton-Montclair line and you may need to transfer to that line somewhere along the way from Penn Station. See the NJ Transit website for info.
Cab fare from JFK would be