Kaş is a long transfer from either airport, which is exactly why it has stayed small.
Dalaman Airport (DLM) is 150 kilometres west, about two and a half hours by road. Antalya Airport (AYT) is 200 kilometres east, about three hours.
Both airports have direct summer flights from the UK and Ireland with Jet2, easyJet, TUI, British Airways and Pegasus; Antalya has more winter services. Choose whichever gives you the better fare and book a private transfer.
Private transfers are the easiest option and most hotels can arrange one. Havaş shuttles from Dalaman go to Fethiye, from where coaches continue to Kaş; from Antalya, coaches run along the coast via Finike.
The coast road from Fethiye through Kalkan to Kaş is spectacular. Coaches link Kaş with Fethiye, Antalya and Olympos several times a day.
Check entry rules for UK and Irish passports, read getting around Turkey and compare car hire in Kaş. Then pick where to stay.