Bangor Airport is the closest major airport. It's about an hour's drive from Mt. Desert Island.
BGR status (courtesy FAA)
FlyHere.com direct flights to BGR
Bar Harbor is the closest airport. It's not in Bar Harbor, nor even on Mount Desert Island, but about ten minutes away, by car, on the mainland.
Bar Harbor Airport is very small. To fly in, you often have to change to a smaller plane at Boston Logan or Bangor International. Also, tickets may be more expensive than for BGR.
There is a free seasonal shuttle bus, Island Explorer bus line 1, from the airport to Bar Harbor. From there you can transfer to free seasonal buses to all points in Acadia National Park. These free buses also stop at the ferry docks serving the Cranberry Isles.