LaGuardia Airport (LGA)
Take the M60+ bus to 125th Street in Manhattan, then take the Metro North train on the Hudson line from Harlem-125th Street station to Poughkeepsie station.
Newark Liberty International Airport (EWR)
Take the New Jersey Transit train from Newark Liberty International Airport to Penn Station, then walk to Moynihan Station (attached to Penn Station), and take the Amtrak train on the Empire Service line to Poughkeepsie.
John F. Kennedy International Airport (JFK)
Two options: 1: Take the AirTrain to Jamaica, then the Long Island Railroad (LIRR) from Jamaica to Grand Central Terminal, then take the Metro North on the Hudson Line to Poughkeepsie. 2: Take the AirTrain to Jamaica, then the E subway line to 34th Street-Penn Station, then walk through to Moynihan Station (attached to Penn Station), then take the Amtrak Empire Service Line to Poughkeepsie.
Albany International Airport (ALB)
We've never done this, but the train/bus situation leaving the airport itself seems like more of a hassle than NYC airports, so here we'd recommend you Uber from the airport to the Albany-Rensselaer Amtrak Station, and then take the Amtrak on the Lake Short Limited line to Poughkeepsie station.