JFK was Eastern's territory, both Eastern and Continental were sister carriers in the Texas air corporation. The most significant route CO had at JFK was a JFK-HNL-SYD flight from '87-'88, it was a DC-10 that operated out of the Eastern terminal. Outside that flight and maybe a flight or two to IAH and DEN nothing substanstial, when the strike hit Eastern CO was already well established at EWR and had no desire to build up JFK.
When Eastern started rebuilding after the '89 stike they focused on LGA and EWR, they even had flights from Islip to MCO, FLL, and MIA. Since Eastern ended their JFK operations around '89 CO also ended their JFK flights and did not return until the late Ninties. However they've maintaned a large cargo operation at JFK all along, and they've even opened a new cargo facility at JFK. The cargo operation at JFK is set up to transfer (by trucks) the cargo to EWR, where they are flown. You might see their trucks traveling from JFK to EWR, you can't miss them they have the huge CO logo on the side.
Here's an article about CO's new cargo facility at JFK,