If I were the sort of person who carried cash, I might leave it in the barn for the farrier. I left checks for years–many of them blank–but there was a place to do so (bulletin board). Some boarders at my barn still do this. I do not, but only because I have started holding my horses for him instead of asking the barn staff to do it.
I think there is an expected level of trust with barn staff/fellow boarders. At least I have one–but I have also been at this particular barn for decades now. I MIGHT be concerned about the lesson kids, I suppose…but only because there are now so many of them that I never meet.
I don’t think I would hold the BO responsible if my cash (or check) went missing, though. It is a known risk, for one. And, for another, things blow away, get knocked off boards, etc in barns. IF there were a pattern of thefts, then yes, I would expect the BO to step in somehow (or at least be apologetic). But if one check/cash went missing randomly? I would chalk it up to bad luck and figure out a safer way to make sure the farrier gets their money next time.