I don't know much about asthma but I have looked into air purifiers a little. My wife is allergic our cats.
Most allergens come from outside the house.
Get a higher quality air filter for your furnace and you should change them more frequently (monthly instead of every 3 months).
Vaccum frequently, get a vaccum with a HEPA filter.
The air purifiers your asking about are HEPA air filters. You buy them based on the size of the room. They cost about $100-$500 depending on room size and they have two filters in them. The charcol prefilter needs to be changed about every 3 months and the HEPA filter needs to be changed every 1-3 years.
There are whole house HEPA filters you can buy that attach to your furnace but I'm not sure if they can only be attached to new furnaces.
Wrap pillows, mattress in a special cover that keeps dust mites out.
You can get most of these things at Target or there are places like
http://www.priorities.com/It gets costly very fast, but it's difficult to put a price on good health. Start your focus on the rooms they spend most of their time in like bedroom, living room etc...