One definition of a good ride: keeping the horse between you and the ground.
How to have a good ride: keep one leg on each side of the horse and your mind in the middle.
All riders, no matter how experienced, sometimes fail to have rides that comply with the definitions set out above. Every rider falls off, sometime. But I think there's some things riders can do to reduce the odds of falling off, and to reduce the odds of injury if they do fall off. Add your thoughts in the comments.
1. Stay relaxed. If your body is tense or you have braces in your legs, arms or back, you're more likely to come off if your horse makes a sudden move. If muscles need to contract to allow you to stay on, and other opposing muscles have to let go before those muscles can contract, you've lost valuable milliseconds of response time. Also, being braced or tense doesn't allow you to easily follow the motion of the horse and tends to push you up out of the saddle - if you're bouncing around there are braces somewhere in your body.
2. Ride in a balanced position. If the horse were suddenly removed, and you were placed on the ground in the same position you have in the saddle, would you stand there nicely balanced or would you tip forwards or backwards or to one side? If you're not in a balanced position, you're vulnerable.
3. Keep riding. Don't let your attention wander - when you're riding, you're riding - keep giving your horse direction and leadership - a horse who has to take over the controls because you aren't paying attention can get you in trouble. And if something happens, don't stop riding - if you stop riding you're much more likely to come off, and if you scream or assume the fetal position, you almost certainly are going to come off.
4. Learn how to do a one-rein stop. It doesn't work in every circumstance, and sometimes you can't use it due to the terrain, but it's a useful tool.
5. If your horse is a habitual bolter, rearer or bucker, figure out what the problem is and fix it. First rule out all possible physical/pain issues and only then address it as a training issue. Get help if you need it.
6. Have an independent seat, legs and hands. A corollary of this is to be physically fit enough to do the riding activities you do. Riding bareback is a great builder of balance, as is riding without stirrups. If you can find a horse that lunges well and someone to lunge you, ride at all gaits without reins or stirrups. And no one should be jumping a horse, even over Xs, unless they've got an independent seat, legs and hands and are able to maintain a balanced position - it always amazes me how many photos there are of kids jumping, even in leading horse magazines, where the kid's leg is swinging back and they have no base of support except their hands on the horse's neck. Too many trainers in the hunter/jumper world give in to the pressure from parents and kids and let kids jump too soon and jump too high.
7. Ride "in" the horse, not on the horse. Imagine that you and the horse are a single body, and ride from your combined center of gravity - sink into the horse.
8. Assess your horse's mental and emotional condition and work accordingly. If your horse needs lungeing or turnout or groundwork, make sure you do it. Pay attention as you ride - there are typically lots of warning signals before things go sideways. Don't hesitate to modify what you planned to do if your horse isn't in the right mental or emotional place to do it. Don't wait for something to happen and then react - get ahead of things and give your horse direction before something goes wrong - keeping the horse's mind and body engaged will help in many circumstances.
9. Don't jump a horse who has unsafe jumping form. One of the most dangerous types of falls is a rotational fall where the horse catches a front leg, or both front legs, while jumping and flips over - people get seriously injured or even killed in falls like these. I've seen lots of hunters, jumpers and even eventing horses with poor jumping form - crooked, or with front forearms that aren't parallel to the ground but instead with one or more knees below the horizontal. These horses are accidents waiting to happen.
10. Don't ride horses you shouldn't or compete in events you shouldn't. If you're a beginning rider, stay away from "Fireball". Don't let your trainer talk you into buying that fancy horse that you're not really experienced enough to ride - many trainers (not all) do this, perhaps because it ensures lifetime employment. If you're an intermediate rider, don't compete in advanced events. This isn't to say you shouldn't stretch yourself and advance your riding skills by doing things and riding horses that are a challenge, but use some sense. If you've got the wrong horse, or if you're scared of your horse, get the issues fixed (physical/pain issues to be ruled out first) and if they can't be fixed seriously consider getting another horse.
11. Make sure your horse is fit enough to do the activities/events you are doing. Unfit horses, or horses that become fatigued, can injure themselves and also their riders. Pay attention to the footing - horses can slip and fall and it's up to you to choose a safe speed.
12. On the trail, ride with safe and considerate companions. Make sure everyone's aware and plays by the same rules. If you're not interested in galloping on the trail, don't ride with people who take off at a gallop without telling you first.
13. And, if you do fall, and you will, no matter how good a rider you are - freak accidents can happen - make sure you're wearing an approved, properly fitting, riding helmet. You can still be injured or even killed while wearing a helmet, but a helmet that's properly fitted and worn can save you from injury, concussion, permanent brain damage or death. I don't care if it's unpopular in your discipline or if it messes up your hair or makes you look like a sissy. I'd rather look like a sissy than be in a coma.