What you can do to help may depend on what the asthma triggers are.
My son was diagnosed with asthma, when he was 2. Most of his issues, so far, occur when he is sick. Since, he is now just 3, we still need more experience on what his triggers are. Our current strategy is to do breathing treatments using preventative/maintenance asthma medicine once a day in the winter (sick season) and bump it up to 2-3 times a day at the first sign of illness. We do fast-acting treatments as needed, which isn't often if we keep up on the preventative/maintenance medicine correctly. The breathing treatments have made a huge difference. Before he was diagnosed and being treated, he would have a terrible lingering cough each time he got a cold or ear infection. Now, it is significantly reduced and his colds don't seem that bad anymore.
Since it's not uncommon for asthma to be allergy triggered, especially with dust mites, we removed the carpet from his bedroom and put wood shutters in instead of cloth shades. We have one of those mattress covers that are good for allergies. Our only current potential issue is that he likes to sleep on a pillow pet, which cannot be covered by an allergen reducing pillow case. We just have to make sure his pillow pet gets washed and dried regularly.
Cold weather can be a trigger for some people with asthma. So maybe keep him inside if it's cold out and he is sick?
If nasal congestion is causing him to cough a lot, maybe look into saline nasal treatments to help keep his nose clear.
Ask your doctor if a humidifier will help. I have mild asthma and my mom used to use one when I got sick as a child. I don't particularly remember it helping that much. You have to be careful with using a humidifier though that it's kept well clean and doesn't get moldy (which could actually be worse for asthma).
I'd also ask your doctor if she would prescribe breathing treatments to your son when he has a cold if he shows asthma symptoms. I've known other parents to give breathing treatments to their kids for certain illnesses even though they are not diagnosed with asthma.
If exercise makes it worse when he is sick, then switch to calmer activities. That can be tough to do with a little one though.
I'd ask your doctor what treatments she would suggest even without the official asthma diagnosis. And if the non breathing treatment options don't work, I'd ask your doctor about that. If nothing else is working and your son is short of breath or wheezing, then it's definitely worth checking out.
Even adults sometimes temporarily go on asthma medicine for some illnesses, like bronchitis. You don't have to be diagnosed with asthma in order for the medicine to be beneficial under certain circumstances.
You could also go see an asthma specialist.
BTW, breathing treatments can be one of two things depending on how well your child can control his inhalation. An inhalar requires the ability to slowly inhale, but it's quick to administer. A nebulizer is typically used for small kids where controlling inhalation (or monitoring their inhalation) may be difficult. The downside is that it takes longer to administer.