Risk versus reward...
To me, it's all about risk versus reward. I can't see taking a young child (under 6) on the bike unless it's around the neighborhood for just a couple of minutes AND they're sufficiently protected. Young children are especially fragile. Even up to 10 or so I have to wonder if they have the physical strength and mental discipline to hold on if one has to take evasive action on the road. Maybe they do... but is the risk worth it?
Most of you have been riding a lot longer than I have, but I've been alive as long or longer than most of you. What most people thought was acceptable risk when I was young is far more than most of us will accept now... at least in the developed world. I was in India last week; the craziness I saw in traffic, on motorcycles, was unbelievable. I had a man pull up to me with his three-year-old son riding on the tank... in a cotton outfit with no shoes and no helmet or eye protection... on a main arterial in a major Indian city during rush hour. If ANYTHING goes wrong, that child is dead.
My sense of acceptable risk certainly has changed since I've gotten older; some of the things I did as a teenager or young adult seem unbelievably stupid now. It's good that God looks after fools.
I've been riding as an adult for a couple of years now, but I still won't take my 10-year-old son for a ride on my bike. However, I did buy him his own XR70 for his birthday, and we took the MSF Dirt Bike class together. Like most children, his common sense can be lacking at times, and he is still developing the ability to think ahead a little. As a parent, one of the hardest things for me to accept is that he will have to make many of his own mistakes... and my job is to try to minimize the consequences of those mistakes because I can't prevent them.
I can accept the risk of having my son ride, with protective gear, in off-road situations (trails, etc., but not racing). I feel that I can control the risk so that it is acceptable. On the street is another matter. You can do everything right, and still get killed by some idiot who isn't paying attention (my only serious injury/accident was caused by that). I'm hoping that riding off-road will expose my son to motorcycling in a safer, more controlled environment... and I hope he DOESN'T get a street motorcycle until he's past his 20s.
I guess it all comes down to looking at the worst that can reasonably happen, and then asking yourself, "Is it worth it?"