Koloa is served by Lihue Airport (LIH) on Kauai, about 30 minutes drive west. Lihue has direct flights from Honolulu (30 minutes, very frequent), and seasonal or direct flights from several mainland US cities including Los Angeles, San Francisco, Seattle, and Portland — check availability, as routes change. From Honolulu, inter-island flights run throughout the day on Hawaiian Airlines and Southwest.
There is no public transport worth relying on from the airport to Koloa. Car rental is the practical approach — pick up at Lihue and you'll have access not just to Koloa and Poipū but to the rest of the island as well. The drive down from Lihue is pleasant, passing through sugarcane fields (now converted to other crops) and the old plantation town of Puhi.