Farmlifeusa, this dust that's coming out of the bale are spores from the mold forming in the bale. There would also be microbial degradation of the plant material, primarily degradation of protein in the alfalfa itself, taking place inside the bale. So the mold and the microbial activity in those bales mean spoilage, which also means that whoever you bought the bale from baled it up too soon, my guess is around 25% moisture or higher instead of the prefered 20% moisture content (or lower, can be down to 15% moisture).
What you should do is NOT buy from the same owner again! There really isn't much you can do since the browning reaction has started (or been going since the last 2 weeks). BUT, you can get some hay preservatives to slow down or stop further microbial action, like ammonia, or organic acids like proprionic or acetic acid. But the application of these chemicals are usually done when the hay is being baled, but you can pump ammonia into the bale to lower the pH and slow down the microbes.
Another thing you can do is design some method of drying these bales ASAP, otherwise the bales are going to loose they're nice green colour and turn into an ugly brown, dusty mess.
Like I said before, the fella (or lady

) you got these bales of alfalfa from didn't leave them long enough in the field to dry in their swaths (lemme guess...didn't bother to feel underneath to see if it was still wet and thought it was dry enough by feeling the top layer), or the folks didn't have "proper" equipment to speed the drying process, or....the possibilities are endless.
Anyway, sorry for rambling off on ye like that. But that's my two cents.