I'm fairly sure you don't ever hold on with just your knees in a caterpillar climb. You always have 1 hand on as well. It's basically caterpillar up, straight edge, caterpillar up, straight edge. Just trying to visualise but I think you slide your bottom hand up, use that to hold on to the pole while you let go with your elbow and bend your knees to pull yourself up the pole, then regrasp with your elbow. Then you release your legs into a straight edge, clamp them around the pole and repeat. (I hope that's right, but if not I'm sure someone will correct. I'm just trying to remember because I have too many pulled muscles to actually do it right now. But I know you never grip with just your legs.)
You want to have a fairly solid elbow grip straight edge, because you're going to be doing this at increasing heights, where you can't just cartwheel down if you overbalance.
I fall all the time and with one exception haven't been hurt. I fall alot because I don't feel much fear, I know how to fall, I push myself too hard and I have very slippery hands (and possibly because I lack common sense). The one time I hurt myself is the first time I tried a Rainbow, and learned the hard way that you probably shouldn't try to transition to a basic invert from there. I basically lost all contact with the pole and was slipping head first. Luckily I have enough falling experience to do a bit of damage control, and managed to protect my head by landing hands first and then coming onto my knees. It would have been fine, just bruised knees if I hadn't landed with the top of my tibia right on the corner of the base of the pole. And also if I didn't ignore the injury and keep poling. Turned out to be a chipped bone.
I think the best thing to do is figure out exactly why you fell, and make sure you don't do that same thing again. Then, you have to try the thing you fell on again, but be more careful if need be. So in your case I would practice some invert moves you know you can do, and help build your confidence. Then work on having a good, confident caterpillar and straight edge. Make sure you know you're using the right technique, and when you feel confident you can try it again, with mats and spotters or absolute silence, or whatever will help you feel confident.
As for preventing falls, always figure out your next points of contact and test them tentatively before relying on them. In other words, add additional points of contact before you take some away. Even holly drops and superman drops can be done this way in the early stages. Extra contact points mean that if you get tangled, or confused, or slip you have as much contact with the pole as possible. Worst case scenario is pole burn. If you do feel you're about to slip, just grip everything as tight as you can, and stay still for a second to figure out how you can get down.
Practice sliding down a pole in pose, maybe peeling off one limb and moving it down an inch, then doing the same with the next limb all the way to the floor. It will build strength and aerial awareness too. It also got me out of a sticky situation once when my foot got trapped somehow and I got stuck upside down on a spinning pole going way too fast 2 meters above a hard wood floor. I kind of crawled down the pole inch by inch until my shoulders touched the floor. And people laughed. Which was probably fair.
