Sean, don't have kids but have a niece that rides with me. Knowing that she would sooner or later we started early enough that she'd never really have the chance to be scared of it.
She was strapped to the sissy bar/backrest for her first ride before she was a year and a half. First rides were only in a grandma's large yard going round and round round (because she didn't want to stop) or later on bike trails in a rural park. Just before she was three she was riding on the street though.
She's especially liked getting picked up at pre-school. Little friends ooh and ahh and ask if they can ride and she always says no. When she was much smaller her then still limited vocabulary contracted to three words when she was riding. It was "oooooohhhh," "more" and "faster."
Now she's so comfortable she's just chattering away all the time back there. She's about five and a half now. Have had occasion to give neighbor kids or others rides and there's a lot more apprehension with their first rides; no matter how much they want to do it.
Important things to be aware of are the obvious (feet position and leg length vis-a-vis pipes, bag lids, etc) and the less obvious (if they are comfortable back there they will go to sleep on you). Obviously you've got Mom to weigh in the decision. Once she gives the green light, however, the earlier the familiarity can begin the better off they'll be.