Off of the top of my head, here's what I can think of:
-can I use my own farrier/vet, or do you have a "preferred" one?
-who is responsible for worming my horse (either you or the BO)
-what are the barn hours
-what are the barn rules
-do you stable overnight horses (can be riskier because traveling horses often are sicker than horses who stay in one place)
-how many boarders are here
-what kind of turnover rate do you have with boarders
-would someone here be able to help me with training/lessons on my horse
-how often are waterbuckets cleaned (and look at a few!)
-what kind of turnout will my horse have
-how long will my horse be turned out
-how many horses will my horse be out with
-are their trailering services available (in case of emergency)
-what type of feed will my horse be eating? What type of hay?
-how often are the horses fed?
-will you feed supplements? Additional cost?
-are fans provided in the summer? Additional cost?
-are blankets put on in the winter? Additional cost?
I also take a look at how clean the barn is...i prefer barns that do not have aisle clutter...and where everything has its place. :)
Hope these helped