You don't mention whether you have a family history of allergies and asthma, and we're not physicians, so continue to discuss this situation with your doctor.
That said, The Money Pit's indoor air quality expert, Jeff May, doesn't recommend that people with mold allergies or sensitivities live in houses with hot-air heat, central air conditioning, and wall-to-wall carpeting. If you do have this type of HVAC system, the least you should do is install a good quality electronic air cleaner to keep the air as clean as possible.
The presence of mold in your house could greatly impact your son's allergies or asthma. If your son's symptoms improve when he is out of the house for an extended period of time, such as a family vacation, we'd recommend that you have an air quality professional do some air and dust sampling for you, to see if mold is present.
If mold is found, removing the mold alone will not solve this problem. You could spend thousands on remediation and unless the source of moisture is eliminated, the problem will reoccur. Mold testing could help narrow down the cause of the mold. Hiring someone who understands buildings before you do a mold test, such as a home inspector who is trained in mold and a member of the American Society of Home Inspectors (ASHI), can help you find the problem and change the conditions, so the mold won't return.