Sherry, I fully understand your concern and this is something WikiTree is very concerned about allowing members to protect sensitive information from public view.
You can set the privacy level of the profile as private and choose family tree to be public or not, while keeping biography private.
The private portions are only accessible by someone on a profile's trusted list, so in order to make it accessible to your family members, they would need to join WikiTree so that you can add them to the trusted list.