Parenting in the pre-online days was probably a lot easier: in general you know who's talking to your kids: they're probably playing in the {street,park,diner,mall} with their friends, or they're in their room listening to music or playing on {game console,PC} which are offline.

Now, kids have phones and Internet-connected devices and the whole world can reach them 24/7, and that includes a lot of creeps.