I'm perhaps over cautious, but I slow down and move as wide as I can. Offering a cheery 'hello' as I approach, making sure I get eye contact from the rider as I slowly pass when it's safe, watching the horse all the time. I've had a little experience horse riding, and I think some non-riders under-estimate how difficult it can be when a horse spooks.
Personally, I much prefer someone to talk to me instead of dinging a bell. Some I ride with ding the bell like crazy at people walking on single track roads, they mean it in a well mannered way, but it does make me cringe.