There's no NJ Transit train or bus service to Mendham so Morristown would be your closest destination. You can take the bus from Port Authority Bus Terminal or train from Penn Station. Both are reached from JFK by NYAS bus ($15) or Aitrain>Jamaica station>E subway>42nd St./Port Authority ($7.25). You can also take the LIRR (faster than subway) from Jamaica station to Penn Station ($13). Check the NJ Transit site for bus/train schedules. Note: some NJ Transit buses/trains may require a transfer along the way.
There's no NJ Transit train