One thing that works for me is to think of the worst-case scenario for any given activity, and what is likely to happen. For example: using a knife to chop veggies. My 9 year old likes to do this. Okay, he could chop off a finger (unlikely) While it would be a miserable thing to deal with, my grandpa is missing a few fingers and it never stopped him from doing anything. He could give himself a bad cut (an eventual certainty) I know how to deal with cuts, I can clean him up and bandage it, or take him for stitches if he needs them, and he'll definitely be a lot more careful next time. He could feel good about himself for being trusted with a responsibility (likely) and chose to eat more of the veggies that night,because he prepared them (very likely) He'll be learning a skill that he'll need as an adult. So yeah, I let him use a largish knife... and sometimes that seems against my better judgement... but I rehash how to be safe each time, and make sure he's actually being careful.
Keeping them safe is only one aspect of being a parent... they have to grow up to be self-reliant adults, and in order for them to do that, they need to take some risks. What and when to allow them to try new things is going to vary kid by kid, but you know best... if you aren't sure about something try and break it down into a slightly easier task and see how they handle it. In the above example, when he started I'd cut a cucumber longways first, so it had a flat side & wouldn't roll around while he was cutting, then he could cut them totally on his own, then we did the same with carrots. I told him since carrots are harder he'd need to use more pressure, so it was extra important to make sure his fingers were out of the way... stuff like that. You can let them walk ahead a bit and practice crossing the street 'by them self' or let them walk into the bathroom alone, and go in 2 minutes later (or with a boy, go to the bathroom door and say loudly, so anyone in the mens room can hear "I'll be right here at the door!") There are ways to take baby steps.
The end result is a more capable person who, when the unexpected happens, like they get lost in the grocery store or something, they will handle themselves better because they've had some practice they can apply to the situation (they'll look around them, like when they cross the street, and know that you are nearby, like when they go to into the public bathroom, and they'll remember what you've told them about if they got separated from you and follow your instructions) Kids need to be able to deal with life.
