I would move your horse.
It's probably not a big deal. Most horses dont NEED to be blanketed. But it worries me that they went out and took the blanket off of your horse after you sent them an email saying it was your responsibility. It is your horse. Like JDI said, most blankets these days are water proof. If they aren't you could water proof them. Unless your horse is in danger the BO has no right to do that that I know of. Now, if you signed the boarding contract and it said this, then thats a different story but it doesn't sound like that is the case.
I blanket my horse. He is at the bottom of the herd and the other horses do not let him in the shelter. It keeps my peace of mind to know that he has a blanket.
If you are saying you will take the blankets on and off then the BO should not be able to go out there and take off your blanket.
I would move just because I think that is disrespectful.
However if you want to stay, it may sound like you will have to sacrifice blanketing.