it all depends how many horses you have, and how much work you want to put in them, but if you have the time, soak the hay in a barrel of water for a few hours, then let it drip out in a net, or a barrel with holes in it. the dust will be gone and hopefully your problem will be solved.
If your horses are in a stable, and you're having a very moist winter, this may also be the problem. make sure your stables are dry at all times, moist air gives them what we call "stable cough" in Belgium. I have the same problem every year with one of my mares. Talk to your vet for some respiratory powder. (usally white or slightly yellow) it does miracles.