A great percentage of serious crimes (from rape to fraud) are committed by family and friends of the victims. Should we not leave our children with our family alone?
The best move is to teach your children how to not be victimized. It is part of "being responsible for yourself". My parents taught me how to be safe in a bad neighborhood because sometimes you have to go there. They taught me how to pick good friends who wouldn't do bad things to me. They taught me how to spot the precursors to bad things. They let me hang out unsupervised. Because they taught me how to be responsible for myself. Why not teach your kids how to navigate the internet safely.