There are three invertebrates known to be predators of Aiptasia, and two fish. Your choice of control will largely be decided on the type of reef tank you have, and the risks you are likely to take by introducing something that could potentially consume more than just Aiptasia.
Hermit crabs: The unsung hero of saltwater tanks, and surprisingly they have an appetite for Aiptasia. They are the safe choice, as they will only continue eating algae and detritus once the Aiptasia are gone. However, like all animals in a reef tank, they have their own tastes. You may be unlucky and introduce hermit crabs who would rather not touch the Aiptasia.
Peppermint Shrimp: Probably the best known and easiest to obtain predator of Aiptasia. Peppermint shrimp can be confused with similar looking that that take no notice in Aiptasia. Therefore, it is important you familiarise yourself with the shrimp so you know what you are looking for, and have the latin name to hand (L. wurdemanni). Peppermint shrimp literally pummel the Aiptasia to death, and can be very interesting to watch. It got rid of my infestation years ago using these shrimp. Be aware that they can also be partial to zoanthids when Aiptasia run out, so it is important to keep them well nourished.
Berghia Nudibranch: This is an Aiptasia eating sea slug and is a natural predator of Aiptasia. The nudibranch has no interest in anything else in your reef tank. It is completely reef safe.To really make a dent on your Aiptasia population, it is suggested to add 4 or 5 at the same time and watch them go to work. As they consume the Aiptasia their gills turn a dark bluey purple. The best place to purchase nudibranch is here.