I don't board but I have returned moldy hay to the seller. One year I bought 8 large round bales of seemingly good quality hay. After it was broken open, it had the white mold that you speak of, all the way through. I called him and made him take it all back and bring me new hay. My horses are fairly easy keepers and they would not eat it. If the owner of the barn doesn't have the smarts or balls to get good hay, I would move.
"Until one has loved an animal, part of one's soul remains unawakened..."
- Anatole France