First off, cats will make asthma worse. Period, end of story. There are supposedly a few breeds that "don't have dander", but my daughter (who has terrible allergies) is allergic to almost all cats. My niece has cats and dogs, and the poor thing now has an almost permanent cough. Dust, pets, some houseplants, some of the more heavily perfumed candles, plug-in scents, all of these things can trigger asthma.
The most important thing is to get two, maybe three opinions on what exactly the form of asthma is. One of them might just have bad allergies, one of them may be triggered by exertion, there are many, many different things but pets are never a good idea, and I have NEVER heard of pets making asthma better!!