A childhood history of asthma does not preclude serving in any branch of the U.S. Armed Forces.
A recent history of asthma is a horse of a different color.
There is a very good reason to preclude asthmatics from combat operations. In Vietnam, for example CS gas was routinely used under circumstances that troops would have significant exposures. In the case of an asthmatic this could cause death.
As bsml stated, the most important thing is never ever to hide or falsify information on your medical history.
Depending on the service there is a protocol to issue medical waivers to those in your son's situation. He has to go through the physical examination and then, upon rejection, apply to a medical board for reconsideration. In the meantime get copies of his medical records.
This question in various forms comes up frequently on another asthma board I am on.
It sounds like his asthma is well controlled for now. As a part of his physical, they should do pulmonary function testing. From what you say, he should pass with no problems. Just make sure that he doesn't try to hide his medical history. They last person that posed the question had very active asthma and was looking alternative treatments. No way they would pass the exam.
Thank you and your son so much for you willingness to sacrifice and defend your country.