Mold exposure can be very dangerous to children. The mold spores that lead to the health issues you've described are pretty speedy, stealthy airborne travelers, and can take up residence in areas of a home that aren't easy to spot.
Improper ventilation can lead to the damp environment that mold loves best, and this could be a primary issue in your apartment. However, an air quality professional will be able to give you the best and most thorough assessment of the conditions and what can be done.
Before finding a local pro through the Indoor Air Quality Association, make sure you've carefully documented any mold sightings and issues, as well as your son's pattern of symptoms. Then re-address the situation with your landlord, who has a duty to keep the property in a reasonable state of repair and safety.
If he or she still resists addressing the problem, be prepared to foot the bill for an air quality inspection, and to move or file suit if the results reveal mold issues that the landlord refuses to correct. Regardless, be sure that you have expressed your concerns to the landlord clearly and in writing. Should the landlord still not take action and the mold problem gets worse, the landlord could be responsible for not only the cost of cleanup, but even the cost of any medical treatments that exposure to mold may cause.