Mostly I look at the health of the other horses. Indoor rings don't matter to me to be honest. I look for a place that will give my horse the best care possible for the money I pay.
I like 12x12 box stalls, a non-leaky barn roof & someone that lives on the property.
Some people don't think its important to have someone living on the property, but after my sisters horse got injured one night and stayed that way well into the morning before someone noticed him - and he had to be put down - I think having someone around could save a horses life.
I look for a place that offers turnout and feeding 24/7 as well as a stall at night. I also like an indoor arena, and tools around the barn you are allowed to use (lunge lines, lead ropes, saddle pads, tack & other equipment) I also like a personal tacks dorm to keep my tack in so other people don't or abuse it,
I also look for healthy horses that aren't head shy or show signs of abuse. The worst thing is having someone turn out your horse or bring your horse at night in that jerks his head all over the place making your horse scared to death of being caught or led.