It's possible to get a bus or taxi to the train station. Distance aprox 10 miles
You could taxi or uber to the beacon train station which has metro north which goes into the city. You could a bit farther to the Poughkeepsie train station which has Amtrak service to farther destinations.
It's best to use a taxi
Taxi is best.
It is very difficult. Especially with regard to where you are going. There is no train from this airport. And the one nearby has limited destinations
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