Children's fears often begin when they can imagine what lay behind the closet door, what's causing the wind to blow or emitting the strange noise under the dresser. And many adults have fears as well (although they may be loathe to admit it). I can still remember my panic from decades ago, when my parents dropped me off at school and I ran behind their truck to stop them from leaving. So I've lived with some form of fear all of my life. But it never stopped me from taking risks, learning new skills and seeking new experiences. The reason? Nobody accommodated my fears. I had to learn how to handle them on my own.
I'm not suggesting you "abandon" your children and cause them unnecessary suffering. On the other hand, when you try to protect them from normal levels of childhood anxiety, you are actually reinforcing their fears and feelings of helplessness. Learn to let go. Calm down your own fears so that you demonstrate the courage and resilience you want to see in your children.