There is no direct public transport from Kalkan to the airports but it is possible to travel by bus with several changes en route. If you plan to travel by public transport to and from an airportyou will need to plan it, as services vary at different times of the year and day.
Whether you are arriving at Dalaman or Antalya airport, you will first need to get the airport bus to the nearest main bus terminal.
For Dalaman, this is the Fethiye bus station, and for Antalya, this is the main city bus station. From Dalaman there is a Havaş company service bus that links in to Turkish Airline flight times which can be taken by othe travellers. From there you will need to get on the local service bus. These small buses stop at many villages on route and typical journey times are 3 hours from Dalaman and 5 hours from Antalya.
There are also some shared Minibus shuttle services to and from Dalaman which on the outward journey start in Kaş and pick up from a couple of points on the main D400 road rather than your accommodation. They offer budget friendly prices but again depend on demand at specific times of day linked to main flight arrivals times and places must be booked.