Kids fall. Its part of the game. I always tell my kids that when they fall off it means they are a real rider now. They get a laugh out of it. I had one kid fall and get drug about 20 feet in the arena. She jumped up and was great. Then her Mom walked into the arena and she started bawling. There was nothing wrong with her. Just seeing her Mom made it all click.
Don't stress. If you are going to teach you are going to have students fall off. Its not always their fault, its not always the horses fault and its not always your fault. Its just part of the game. We are trusting these huge animals to carry us around. Poop happens. The most quiet of horses will have their moments.
As for the poster who mentioned relying on arenas. I like to keep my kids in an enclosed area until they get an idea of how to control their horse. It just minimizes the degree of the disaster. I actually keep my kids on a lunge line for about 8 - 12 lessons before I let them have their reins and ride on their own, then put them back on the lunge line to learn to canter. You actually are required to teach beginners on the lunge line when you do your coaching certification.
BB ~ 2014 Trakehner Bratty Mare ~ 1993 CSHA Em ~ 2007 Standardbred