Anything that may trigger an allergy attack in an aquarium can be easily controlled--unlike pet dander, which is pretty well present with any pet. Cleaning the surfaces exposed to water regularly, and avoiding materials that will absorb water like wood is easy enough, and part of regular maintenance anyway. I've never had a reaction to my tanks--and know that the humidity provided from the tanks actually helps quite a bit with asthma.
There are many people who have a skin reaction to a variety of fish foods, with blood worms being most prominent. Flake foods that crumble easily to dust can become airborne more readily--but with the variety of foods and packaging, even that shouldn't be a problem.
I'd also point out that a huge number of hospitals and doctors offices have tanks. Just as they don't keep blooming plants due to the allergy potential, I can't see them keeping an aquarium if it was a common trigger for asthma.