Here is a chart that illustrates my thought process in this situation:
Sometimes your child wants your help. If you can help, then why not do it? If you do, he will learn that getting the help he need feels good, and that it's cool to help other people when they need it.
Sometimes, you can't help in the specific way your wants you to. In that case, you and your child may be able to find someone else who is able and willing or figure out a different way to help him accomplish his goal. If you do this, he will learn that you care about his needs, that you are resourceful and creative, and that there is often more than one way to do something.
Sometimes your child doesn't want your help. If it is safe to do so, why not stand back and let him figure things out? You can let him know you are there if he needs you, but stay out of his way. If you do, he will learn that you trust him to know when he needs help and that you have confidence in his abilities. Also, he will have the opportunity to get better at whatever it is he is struggling with.
Sometimes your child doesn't want your help but he is doing something dangerous and doesn't understand the risk involved. In that case, of course you step in if you can. If you are unable to intervene in time, you can help by comforting your hurt child and calling for more help if needed. Once the situation is safe enough, you explain why you stepped in, and help your child understand the danger and figure out a safer way to accomplish his goal. If you do this, he will learn that you are looking out for him.
These are all things I want my children to learn. How about you?
On the other hand, what does a child learn when he wants help and his parent refuses to help him? What does he learn when he doesn't want help but it is forced on him constantly?
I have read a bit about this issue recently, and I think maybe some people are confused about it. I think some people confuse "letting" a child do something on his own with "forcing" the child to do it on his own even when he doesn't want to. There is a big difference. There is also a big difference between insisting upon helping or doing something for a child, and being willing to help when he wants it. The child's wishes should not be ignored in these discussions.
Here are the articles I'm referring to, if you are interested:
From The Atlantic, Why Parents Need To Let Their Kids Fail
And from Alameda Patch, Please Don't Help My Kids