Be very sensitive to their fears. It may seem funny or cute to you, but it is real to the child. Usually your child will grow out of many of the fears (some fears will be lessened and will not go away even as an adult), so do what you can to minimize the fear (ex: do not show scary movies if this causes nightmares, remove all scary books if this causes fear, if a child calls out seeing a bug, respond by removing that bug). Do not dwell long on or talk long about fears with them, unless they insist on it and share it with you. Even then keep it short. Just be consistent and be there for them and don’t make a big deal about it. It is OK to explain briefly that in the darkness they will be safe, but as children they might not be able to help it. If you talk too much about it they may think they are right in their having fear since it is so important to you.
Be aware of the fact that fear may cause poor behaviors. For example your child may disobey until you will "not" let them do some activity, like swimming (which you may do every Saturday) because she may be fearful of swimming (drowning) or not being able to perform well at it. If you are alert to possible fears, you may be able to minimize them. Don’t force children to do something that causes them fear.