If you are referring to an air stone, your
betta should have one. Bettas, being tropical fish, should be in warm water (80 degrees). Warm water supports less oxygen than cool water, and the oxygen lost to the water being warm should be replaced.
Though Bettas do breathe air at the surface, the oxygen is needed to help support the fish, as well as help the bacteria colonies that form the backbone of the cycle do their job.
The air rising from the airstone also helps to break the tension at the water's surface to allow certain harmful gases to escape. But you don't want the aeration device to run at full steam, just enough to give the water a bit of movement. (Bettas don't seem to care much for a lot of motion in the water.)
You'll need an air pump, air line, air volume controller (or gang valve) and whatever stone(s) you want to use. If you have more than one tank you want to feed air to, you might be able to get away with a larger air pump and a gang valve with enough outlets to serve each tank.
Rena makes a pretty quiet pump, which you can get at drsfostersmith.com if you can't get one elsewhere.
Mike