I would never take him somewhere he wasn't welcome, or take him close to horses I'm not familiar with. Should go without saying, but I know people do it. The barn I ride at definitely allows boarders to bring dogs, I actually had the conversation with the manager while she brought up her new German Shepherd puppy to play with another boarder's Vizla. The Vizla spends his entire time there running at high speed from person to person, and no one cares, including the horses. Of course he isn't chasing anything. The manager has two other dogs, the trainer has a pointer puppy. Probably not what most people (myself included) would imagine at a race horse stud farm It's a very relaxed place but all the people and horses seem sensible. There are separate barns for the boarders, stallions, and mares with foals.
He would NOT be allowed loose at the barn, or ever in fields with loose horses, even on a leash. And would only be left alone if I was confident he would be comfortable and well-behaved. He would be allowed to hang out in the tack room, but I'd worry about someone letting him out accidentally. The field fencing is wire mesh (small openings, less than 4"), I suspect to keep out dogs. I wish we had an indoor, especially this winter!
I do also have the possibility of taking him to a friend's small farm if that would be a good "intro level". She has two horses and a collie that's fine with them who could be a mentor.
I guess I'm looking for any specific pointers about how you introduced your dogs to your horses in a way that kept everyone calm.