Thought of a few more . . .
Security? Are people around 24/7 in some barns, no one is, but it's a good thing to have
Health requirements? Coggins, flu/rhino, and equine encephalomyelitis are required, the more vacinations required, the less likely your horse is to get sick. They should also have you prove your horse has had regular dewormings.
Good footing in arenas?
If you live in a climate where the water in the water buckets will freeze, do they have water heaters?