Airborne particles and ammonia fumes in your horse's stall are a concern, but so is the air quality in your indoor arena.
A dusty indoor arena can be as harmful to your horse's respiratory health as the air in the filthiest stall, and possibly more so because a working horse breathes harder and pulls particles deep into his lungs. Initially, he may simply cough a bit, but over the long-term severe lung damage can occur. A horse ridden regularly in a dusty arena can develop heaves years down the road or end up with permanently diminished lung capacity.
It's not difficult to tell if an arena is too dusty. If you need to blow your own nose after a ride and the tissue is filled with footing particles, that's a clear indictor of poor air quality. It doesn't have to be that obvious, though. Wipe the top of a mounting block, chair or ledge in the arena, or clean a wall-mounted arena mirror. Then, monitor how long it takes for dust to accumulate again. Ideally, it will be a week or more before there is an appreciable layer. If you can write your name in the dust after a single ride, the footing in the arena is dusty enough to pose a significant threat to your horse's lungs.
Keep in mind that it's not only your horse's lungs that can be damaged by dusty footing, but your own. Riders and instructors who breathe in particulate-laden air are prone to respiratory illness, not just from the irritation of the dust, but from an overtaxed immune system as their bodies work hard to clear the foreign debris from their lungs.
Dampening arena footing with water may cut down on the dust slightly for a short period of time, but the only long-term solution is to re place the footing. There is a wide select ion of modern, dust-free footings available and they are worth the investment at every price point.