Ngurah Rai International Airport — the only airport on Bali — is about three kilometres south of central Kuta, ten minutes by taxi in good traffic and double that in bad. It handles flights from across Asia and direct services from Australia and the Middle East. From the terminal, use the airport's fixed-price taxi counter; ignore the touts in the arrivals hall.
If you are coming from elsewhere on Bali, taxis and ride-hail (Gojek and Grab, both via app) are the standard option. Public buses do not really serve Kuta; the Perama tourist shuttle network runs scheduled minibuses from Ubud, Sanur, Lovina, and other tourist towns and is the cheapest way over.
From Java, the long-distance buses arrive at Mengwi or Ubung terminals north of Denpasar, with onward taxis into Kuta. Many travellers prefer to take the ferry from Banyuwangi to Gilimanuk in western Bali and then a tourist shuttle south.