Pretty much what Drac said.
Arguably the PS4 is very nominally the better console, but for most practical purposes there’s nothing in it. If they (or you) have friends with consoles then buying in to the same system is worthwhile so you can lend each other games and (don’t knock it till you’ve tried it) play online. For this reason, I’ll be getting an Xbox One.
Is it actually a ‘family’ console though, or just for the kids? If the latter, a Wii U might be a more anklebiter-friendly purchase. It’s hands down the worst of the three but probably the most child-friendly. Or, consider last-gen even, The 360 / PS3 / Wii are still decent machines, will cost a fraction of their big brothers to buy, and considerably less in care and feeding. For the price of a PS4 / One you could get a PS3 / 360 and enough games to keep them happy till the heat death of the universe.
As for playing online, bear in mind that most multiplayer games give you the option of playing in public games or restricting to just friends. Avoiding public games will dodge most of the horror stories you may have heard, factual or otherwise. Playing online with little Timmy at number 27 is no more risky than them playing together offline.
Don’t forget to budget for a second controller.