The Pawling Station about 3 miles south for the trail by road is a good spot. There are small places to eat or buy groceries. Walk 1/2 to NY 22 and find a relatively easy hitch. Multi-day parking is very cheap at the meters in the station parking lot. (you can get a train pass for a bike, eg. from the station, bike to the trail)
Amrak passes near Asheville, NC Bus goes to Asheville. Get a shuttle to the trail.
Busses travel almost hourly from Delaware Gap to NYC. Therefore public transportation can get you to the trail at the NY/PA border and the NY/CT border.