J.S.
It is a dry cough or a wet cough? Does it sound like a seal barking?
Try giving him a honey based cough syrup (they contain less than 1% alcohol) and can be found at a health food store, immune support vitamins, saline nasal spray, cold mist humidifier, and a good balanced healthy diet. Also it is never to early to teach him good hand washing methods.
If he doesnt have a fever then (generally) you are just dealing with a run of the mill cold.