Well, I could be snarky and tell you "get over it."
But in fact, it's perfectly rational. Falling hurts. And it's scary. Sadly, it's going to happen. You're tightrope walking on a slippery surface, after all.
In my experience, however, falling is scariest in the anticipation. Once you slip, it's in the hands of fate. What you need to do, then, is learn how to fall, and more importantly, how to overcome the fear.
The first, and most important thing, is to fall.
I teach this the same way at both ends of the age spectrum--toddlers and adults, who tend to be the most fearful of falling. Sit down, stand up. Sit down, stand up. Sit down, stand up. You get the idea. You want that ice surface to be your friend. If you're skating with a young child, have a race-- who can get down and then back up the fastest (always let the little guy win). Pretty soon, the kids will be throwing themselves to the ice and scrambling back up and having a blast at it.
Adults are really really afraid of falling. And they have good reason to be. Most of the injuries I've seen in non-competitive skaters are adults. They fall poorly, and it's a long way down. They come in with more physical deficits, even the healthy ones. So go with that. With most adults, I leave them in their comfort zone, or just barely over it. (The comfort zone of course is a moving target. Once they get comfortable somewhere, you push the bar ever so slightly.)
If you're really not going to let yourself fall deliberately, you have to do everything else the coach says. My favorite skating mantra is that doing the scary, counter-intuitive thing (bend your knee, lean into the edge, turn your shoulders not your hips) makes you safer. If you fight the design parameters of the equipment you are more likely to fall.
Learn how to fall well
Tuck and roll. You want to hit the soft tissue, not the hard stuff. This means chin to chest, hands to tummy, curl your back, hip to the ice. It's better to fall in motion rather than from standing because a lot of the energy will go into the slide instead of your kneecap. Never ever catch yourself on your hand. Whatever position you are in, roll over when you hit the ice. You can practice this.
Do the scary thing
For adults this might mean crossovers. For kids, it's the jumping. The less you practice a skill, the more likely you are to fall when doing it. But don't attempt skills you're not trained for. If you're just learning a loop, you shouldn't be playing around with axels. If you're unsteady on a one-foot glide, what until it's solid before trying crossovers.
All beginning adults should wear wrist guards. As regular readers of this blog know, I'm a big fan of head protection, in particular the Ice Halo®, basically a "stealth" helmet that just looks like a really attractive hat. Don't wear bulky kneepads--they make you stand funny and get in the way. A simple sleeve with a soft pad over the kneecap is plenty. If you're in hockey skates, especially beginners, you should have a hockey helmet.
Adults are allowed to cry (sort of). Children are not (sort of). As a coach, I need to know when a child is injured and when they're just scared. If I treat every fall like an ambulance-worthy injury, I'm helping to create the fear. If your child falls, observe her for a moment; she's probably watching you for cues-- "am I hurt?" I have a "no crying on the ice" rule-- you can't cry until you step off, then it's okay. If a student cries despite the no crying rule, I know they're hurt.
Adults need to suck it up. If you pretend you're hurt worse than you are in order to save face, I have to go with that. But you're just reinforcing your own fear if you do this. Accept the level of injury--if it's just to your pride, move on. By the end of the lesson, no one will even remember that you fell.
Try a different type of skating
Some people can't get past the fear. But that doesn't mean don't skate. It really just means don't jump. There's still hockey (nice, padded hockey), ice dance, figures, speed skating and moves. There's also just skating-- come to public and skate around. I am exasperated by people who take lessons and then refuse to learn things because they're scared. If you don't want to learn new things, don't waste everyone's time with lessons. Taking lessons implies a desire to learn new stuff. Just come and skate for fun.
I want to go to Nationals, but I'm afraid of falling
Yes, I have had people say this to me. It is a nonsense statement. If you're afraid of falling to the extent that you can't train, you have zero chance as a competitive skater. If you're a competitive skater, or even just going for the tests, You Will Fall. The vast majority of experienced skaters fall well. Training makes falling safer.
It's like anything. If you practice it a lot, you get good at it.