If they really want to participate, Then there should be some safety measures, I. E. A doctor, Emergency gear, Extra money to help pay the hospital bills in case of an anomaly. As long as they can be safe, I see no merit in holding asthmatics back from enjoying their passion.
What triggers an asthma attack in one individual can be entirely different than the trigger for another. There was a time when kids with asthma didn't even have to take PE because they thought it was bad for them. That is changing. Anyone who has the talent and ability to play professional sports is going to be in touch with his or her own body and will know what it can and cannot take. Let the athlete decide.
Yes, asthma sufferers should be allowed to play professional sports, when exercise is a known asthma trigger, because it should be their choice whether they want to play or not. Smoking is known to trigger lung cancer, but we still allow people to smoke. A person can choose how much risk they want to assume.
Professional sports typically have access to high quality medicine, and often the best. If an athlete is known to be an asthma sufferer, his or her team can take precautions and be ready to render assistance if necessary. The condition can be managed to the point that the player would not likely be in jeopardy, and would likely be in better health for doing something they love.
Whether or not asthma sufferers should play sports is a decision better left to a doctor. In the absence of permission from a doctor then I can't see how any other authority figure can give permission for an athlete who suffers to play and sport, even a sport considered less strenuous.