Although chinchillas have no dander some people can be sensitive to some of the products they use.  This article is for persons with high allergy problems or asthma.  

Chinchillas require bedding for their cage, pine is a high allergen product.  Recycled paper bedding is great and safe.  do not use the color dyed type.  This is a dust free and very absorbent product.

Hay is also a high allergen.  Usually the hay dust is the irritant so I suggest using pressed hay cubes to prevent hay dust allergens in the air. 

Dust Bath can lead to air born dust particles.  Reduce dust baths to at least one per week.  Maybe while the cage is getting cleaned for the week your chin can have his bath.  In a dry bath tub place the dust bin .  Let your chin bath in his bath this way.  The dust will settle in your bathtub and you can wash it down the drain.  For the allergic person it's a good idea not to handle the fresh bathed animals for a few hours.