Just like mentioned previously regarding allergies in diving, a general answer cannot be given here, that suffering from asthma makes diving impossible!
Asthmatics have the possibility as well to get to know the underwater world under certain circumstances.
The greatest danger when diving with asthma consists in possibly not being able to react adequately in case of asthma. The compressed, dry and also cold air from the compressed air bottle is hereby the largest risk factor. But also the possible exertion under water, e.g. in case of emerging currents, can also be a cause.
Therefore, affected persons should definitely consult a diving doctor regarding their diving projects!