The nearest airport is probably Sky Park in Red Hook. The problem is that it is almost all general aviation and is privately-owned; you won't fly commercial out of there.
About as far away, right past the Kingston-Rhinecliff Bridge on the other side is the Kingston-Ulster Airport, which is a public-use airport but I don't believe it flies commercial. It's possible, as some other airports around here did at one point, but its website is not that specific.
The closest airport to Rhinebeck that will actually fly you somewhere commercially is Stewart Airport in Newburgh, but I'd say that's a good hour away from here. Albany International Airport to the north is farther away but not that much farther.
