Hi Samn :)
Those steam cleaners are great. My family used them when I was a child and it helped so much! I still use one every so often along with my Hepa Filter Bissel Bagless Vacuum (cheap one from Wal-Mart that caused me to store my Electrolux). My sweeper gets used on the furniture, mattresses, curtains, etc. Whenever used I notice such a difference in the frequency I have to use an inhaler :)
Do not feel bad for the dust. . . It can't be stopped and your son would have developed asthma anyway. It's in his genes. My family used to feel guilt until the lung doc told them the reality of the situation and to be glad it was caught and controlled early on.
Another option is to have a "Roto Rooter" person come and clean you air ducts in you house. And be sure to change your filter in you furnace monthly. That will cut down on the dust a lot. I also buy a can of spray charger that collects particles to spray on my filter. It smell kind of like some sort of citrus :)
Good luck with your son!
Much Love, Hugs, Peace & Comfort :)