Ask around to see if anyone you know has had any experience with them. Check online reviews, and pay special attention to the ones with 3 stars as those tend to be the most accurate.
Definitely do what everyone else is saying and visit the stable to see if it fits your standards. I've found that checking the amount of bedding in a stall and quality/quantity of water are both very telling.
I especially like what Mark said, too. See if you can observe a lesson because that can give you a good feel of the overall atmosphere of the barn.
It is better to visit the stable before putting your horse or starting to ride there. And do a surprise visit, it is better to see the real state of the stables.
To me, the horse has to go outside everyday (be ridden or go to the paddock, walker...). If I cannot come, I would like to be sure someone can take care of my horse.
Also, I want the stable to be clean often (fresh straw) of course.
Every rider has his own criteria but to me, it is important to see if:
- the box is "cleaned" everyday (remove the droppings and add a little bit of fresh straw)
- the horse can go the paddock everyday
- the atmosphere is pleasant
- the price is correct for the services
I recommend you to make a surprise visit to the stable to really see how it looks (horses, box...).
You have to be aware of the straw, if you see the horses are almost on the concrete, you'll already see what kind of stable it is.
Try to go during the feeding hours, to see how they're fed, and how much hay they have for example.
If they give some lessons, try also to listen at what they're saying and if you like the kind of recommandation they give