I'd be careful with steam as so young. I'm asthmatic and steam makes me worse, I find it much harder to breathe in steam rooms or saunas but I'm old enough to know not to go in them, not that I'm suggesting you might take him in either, more a general point that, if he is asthmatic then until you know what might affect him then take care :)
My asthma was a dry cough too though I can get a cold/chesty or productive cough and asthma symptoms at the same time so worth getting him checked if he is wheezy. Mine improved a lot as I got older though it still troubles me sometimes, often when I get a bad cold/chest infection/virus etc...
I got the horrible chesty thing going round recently and ended up on steroids as my breathing got really bad.