You should feel 'pleasantly-tired' at the end of your workout. Slow down or stop. Any sport you want to do is okay as long as you are prepared. Interval training' is training that alternates strenuous and light exercise. Try to avoid any sports that expose you to your allergens, such as grass and pollens.
Swimming is great, but some people have problems with chlorine.
Other good sports include racquet sports, basketball, baseball, some track and field events, volleyball, soccer, or hockey. Long distance running or cycling with continuous hard effort may not be as easy and exercising in cold dry air may be difficult. But with suitable medication and training you can do those too. Don't use asthma as an excuse for not participating. Be realistic as to how much you can do until you are conditioned.