Aiptasia are certainly the most rapid producing nuisance anemones in our captive systems. Left unchecked they can take control of an aquarium in a short time. Their powerful stinging can irritate corals and clams to the point of death. These unwanted anemones usually come as hitchhikers on live rock. Mostly spreading of this species is due to over feeding, and high nutrient levels. In my years of reef keeping I have gone through many outbreak of Aiptasia, after the death of a large fish hidden away in the rocks, or by over feeding with the intent to grow something more rapidly in the system.
You can control the outbreak much easier by limiting the amount of food protein skimming, water exchanges and dosing vodka & a few other methods as well. While this may thwart reproduction it will not completely irradiate the existing anemones. There are manual processes like inject a solution, kalkwasser directly into the base, and even focusable laser pointers. These methods kill the anemone completely and the laser seems to be the most effective (use with caution) These anemones quickly retract into the rock, making injections a bit more difficult unless the specimen is large.
Another method is to introduce biological control, for example an animal that will utilize the Aiptasia as a food source. There are a few on the market like Butterfly fish, the Caribbean Nudibranch, or the Peppermint Shrimp. All of these have been documented to eat Aiptasia and will move on to other food source once the Aiptasia Anemone has been eradicated. Butterfly fish don’t seem to acclimate very well in the aquarium. They are prone to stress easily then the dreaded ick outbreak. The Nudibranch are a risky proposition, they are not easy to identify and a LFS may end up selling you species that will consume coral as well. I think the best bet is the peppermint shrimp, and laser pen.