I used Phuket Shuttle recently to/from Phuket Airport to Patong Beach. It was 750 THB each way. Reliable service, you can book and pay ahead of time online. A rep meets you at the airport with your name up on a sign so you don't have to worry about trying to find a taxi etc., helpful if you're arriving after a long flight.
From Phuket Airport to Karon Beach appears to be 850 THB for a standard car.
https://www.phuketshuttle.com/