As a lawyer very familiar with torts (but this is not legal advice, just commentary), it’s really impossible to know given what you have presented, and it is probably a grey area where it could go either way, depending on how good your lawyer is and how the judge feels about negligence.
Not worth a lawsuit over the vet bills, IMO.
But if you want the barn to keep the arena open, I wouldn’t recommend it. If there is not money to fix the issue, the BO is very likely to just close off the arena as a place to ride. They might if they even get a whiff people think they are on the hook.
There are some solutions, like boarders banding together to fix it with permission, a 50/50 split, or a board raise and a contract that requires boarders to stay a period of time to ensure the cost is recouped. Most boarders at low-mid facilities don’t want to do this, though.