Honestly - a large amount of crimes in Japan go unreported. Many murders are written off as suicides (which contributes to the high suicide and low murder rates reported by the government...). I don't really trust crime statistics coming from a country that's heavily influenced by Yaks.
This has to do entirely with public perception (as user alexleavitt also mentioned).
In Japan, I feel safe to leave my bag on a bench while I go use a public restroom. I expect my bag to be there when I return. It's probably not any safer, in reality, than it is where I am in the States - but I would never leave my bag on a table unattended where I live. I would expect it to be stolen.
In reality, it would probably be there when I get back. No different than in Japan.
When I walk around Los Angeles at night, I walk defensively and make sure I keep alert at all times and I'm always paying attention to my surroundings. When I was walking around Tokyo at night - it felt no different than the day. Except it was dark outside. I'm certain that had I had better knowledge of the "dangerous" streets of Tokyo, my perception would be different.
If I feel safe, I'm likely to feel my child is safe. In the US, a child walking on the streets in most suburbs is likely just as safe as in Japan... but public perception means you'll get child protection services called on you if you think about letting your 9 year old walk down the block to the store to buy a soda.