OP - I have given this some thought and read the other posts; it seems to me that a driving person is almost always going to be in the minority if you board. So to reinforce my earlier comment and those of others, if you have a good facility with some adequate driving space and a supportive, open barn owner or manager, take advantage of that and go out of your way to encourage the open mind of others. Weigh the pros and cons of a good facility vs. a couple not so great boarders and really understand that these situations are all about trade offs.
We don't get less brave; we get a bigger sense of self-preservation........
I've been in your shoes a lot. My sympathies. Just be as polite and considerate as you can and make sure you stay in communication with the BO. You don't want boarders ganging up on you and turning her ear against you, no matter how supportive you think she is. It can happen.
One great way to despook horses is to let them "chase" the cart. When you're driving, have their rider follow you at the same rate of speed from a safe distance. As the horse starts to tune you out, you slow down and let them come alongside you (again at a safe distance). When that becomes boring, you slow down more and let them pass you so you are now behind them, you walking and them trotting. They will eventually catch up to you and pass you again, and soon you can pick up a trot again no matter where the other horse is. It really doesn't take long before the horses don't even bat an eye - and it helps the RIDERS, too.
Very much enjoying this thread, thanks everyone for constructive suggestions. We have just taken on our first boarder who has a nice, well-broke draft horse and intends to drive. Another boarder with a draftie is now interested in trying it too, who will be sending her horse (who supposedly has driven) to a professional for a refresher course and lessons for herself.
Years ago I had some exposure when there used to be a large pleasure driving show near here, and while occasionally someone under saddle would spook, usually if the rider treated it as no big deal so did the horse.
We'll definitely offer a "meet-and-greet" day for riding boarders so their first experience with a cart is a positive one!
I say definitely do what you enjoy with your horse. If your BO allows you to have a cart and drive, go for it, and don't let others discourage you.
Where I am at there are literally NO driving barns, and it's very very very hard to find a barn that will allow driving carts at all.
My wonderful BO allows me to keep 3 carts! I'm the only one at the barn who drives (well, OK, no someone else just trained their pony to drive too.). Every single horse on the property has seen the carts, and I've never had any incidents.
At the previous barn, I also never had any incidents with the horses (sure, they all get a little snorty and high-tailed when they first see the cart, but they settle down and deal with it). But I did have an owner that threw an absolute fit about the cart (a mini, mind you) being in the barn. She claimed we were scaring her horses (that she never rode, either). She mad our lives hell, but pretty much we just ignored her. The horses never cared about the cart that much.
Where I am at now, I go out on the roads all the time, and pass barns. There is 1 barn I go by that has their outdoor bordering the road, so I do get a little worried when I go by there. If I do see someone riding, I will usually call out. I have startled a few horses as I come up onto the arena, past the barn, when they can't see me coming. But I've never gotten anybody dumped.
Most horses just want a good look and a little run and that's it. Out on the trail, if I see people riding, I do tend to keep a keen distance, just in case. But if the trail gets tight and there's no choice, well, there's no choice. I have had riders turn around or just stop and wait until we go away. I've really never had too much of a problem.
I know another lady who drives and boards at a massive barn. There are 3 drivers at a barn with over 100 riding horses. Talk about outnumbered. It works, though. But the driving horses don't utilize the indoor arena as far as I know. Too busy, and "too scary" for the riding people. Especially kids taking lessons, et al.
Out on the 600 acres of property, though, there isn't much of a problem when riding meets driving.