We have a place for you....
Occasionally police depts. will accept applicants with medical conditions, but they have to be well-controlled for a certain period of time. This is to protect all parties involved: fellow officers, you, and the public. I'm sure you can imagine what would happen if you had an asthma attack at an inopportune time!
Also, I don't know how you would do during training, if you have to do the pepper spray exercise.
I Googled "police officer with asthma" and got several hits. I think it will vary by department and severity of the condition.
Dispatching is fun and exciting. As officers, we handled dispatch duties until our dept decided to civilianize dispatch. I think it is worth considering! There are also other civilian positions within the department, such as crime analyst, applicant processing, etc.
Having developed health problems at the age of 38, I understand how discouraging it can be to have limitations placed upon your aspirations. Try to keep a positive focus, and find ways to go around obstacles. It sounds like you are already doing that by thinking about dispatch. Best of luck to you!
Depending on how bad your asthma is would be the deciding factor. If your asthma is to a point that it can interfere with you ability to defend yourself or others, then you should be a dispatcher. If you asthma is not at such a point and you can work through it with ease then follow your dream and become an officer.