Airport taxis operate 24/ 7 from Denpasar airport, Bali - which, by the way, is nowhere near the actual town of Denpasar, so don't think you'll be smart and spend your first getting-in-late night there. If you are going to Sanur, Kuta, Legian or Seminyak, an airport taxi is a good choice, and pretty cheap.
However if you are going to Ubud, where i live, it's much better to get a driver from Ubud to come down to pick you up - those southern drivers are a-clue-istic - about how to get around Ubud and her environs. If you have already booked your accommodation, ask that place to organise a driver. It may come free but for sure they'll know the way. (Ubud-airport fare is approx $25-35 depending how far out of the town, and how horrendous the traffic; the airport taxi may be $25, but will ask for more once they get lost, or want to drop you off at the main street and get you to find the place yourself.)
I can recommend one of the drivers from my village, Pak Kung. He's very reliable, a village leader. His facebook address is https://www.facebook.com/wee.kung.5. almost 4 years ago