Transport to and from the airport is available for Olympic Airlines passengers.
in the form of Olympic Airlines shuttle buses.(Three Times Daily).
The duration of the above route is about 40 minute
Passengers of all other flights can take a taxi, or walk towards the giant roundabout outside the airport gates and find a KTEL bus running to Kos town, Mastihari, Kardhamena or KefalosBuses.

Taxis are available.
The taxi to/from the airport will cost you approximately 30 Euros (one way) 
The taxi rank numbers: 
+30 22420 23333 
+30 22420 23344 
+30 22420 22777