The barn that I board at offers dude string rides. In fact, one of my best friends is the head wrangler and leads the rides (usually).
Everything that callidorre said is great. There are only a couple of things I would add.
First, have a dress code and make sure people booking a ride know what your dress code is. Let them know that if they come improperly attired, they won't be able to ride. For example, at our barn, if you show up in sandals, don't expect to get on a horse. AGREE. I wish we had this. Shorts are fine to me, but if you have improper shoes that impair you from keeping your feet in the stirrups and your heels down, you shouldn't ride because of safety.
Second, have helmets available in a variety of sizes. Make sure they are clean and disinfected after each ride and I would even go so far as to encourage riders to wear them. Our BO makes anyone under 18 going on a ride wear one. here, everyone under 18 has to wear one (unless you're 16 or 17 and your parent approves of you not wearing one). We have S, M, L, XL helmets disinfected after every use. One suggestion is to make sure you have a lot of M sized helmets- we take rides of up to 8 and it seems like all those rides, everyone wears helmets and they all have M sized heads, though we only have I think 6 M helmets.
Third, make sure the horses you have are suited to their job. I hate to say it, but my BO has a few horses that just aren't suited to being dude string horses. They're either too inexperienced (in my opinion, a horse needs to be dead broke to be on a dude string) or too lazy. I've ridden one of the dude string horses and am not a slouch when it comes to riding. I had to kick the snot out of the horse I was riding to get him to leave off eating grass and finally ended up having to direct rein him in the tom thumb bit he was in to get him to pay attention to me. Definitely a good point. We have good horses, but they can have their days. If we are worried about one not cooperating and trying to eat, we will either walk with them or not rent them out if possible. Most of our horses won't get spooked by anything we see/hear but you know how it can be... Sometimes things can happen. Thankfully if most of our guys were to spook it'd be in place and not taking off. But, in over a year I've seen no issues with this.
Our barn offers several different rides. There's a one hour ride; a two hour ride; a four hour ride (out to the lake and you take a picnic lunch); a ride up to some Native American ruins; what the BO calls the "Hi-Ho Silver Ride," where you get to trot and canter (if you feel confident enough to canter); and there's a moonlight ride available on every full moon during the summer.
Here's the website for the barn I board at: Granite Mountain Stables??
*Funny side note: The three horses in the top left pic on the home page are Bubba (the bay), Reno (the paint...who is also the horse that I rode), and I'm pretty sure that the sorrel is Jack. It might be Skeeter, but I'm almost sure that's Jack. I'm pretty sure that's Jack in the bottom left pic, as well.