Any of the NYC-area airports are options, as well as Albany International Airport, which is serviced by United, JetBlue, American Airlines, Delta, Frontier and Southwest. You can drive from the airport to Kingston or consider the public transportation options below.
Take the Metro North Hudson Line to Poughkeepsie or Amtrak to Rhinecliff and then get a taxi or Uber to Kingston - it's a 20-minute drive from the Rhinecliff station to Kingston, and a 30-minute drive from Poughkeepsie. From Poughkeepsie, you can also take a bus provided by Ulster County Area Transit.
There is a Trailways bus that goes directly from NYC to Kingston, leaving from Port Authority.