From the little I know from human and animal asthma, it's a good idea to get rid of most household cleaning supplies (replace with white vinegar and baking soda) and any other scented things such as candles, perfume, etc. as much as possible. Clay litter is a problem, especially scented stuff, so I would try different litter types. And if it applies and you can afford it, getting rid of carpets is also a good idea.
Open windows as much as possible and at least for 5 minutes every day.
Depending on your heating/air system, an air purifier might or might not make a difference. Clearing the vents at least once a year also helps.
Indoor air quality is affected by all of these things. Dust isn't necessarily the worst part.