Japanese will do bad things when they are sure they can get away with them. You won't get away with stealing. There are police everywhere (in a good way to be honest), and they will catch a thief.
The claim there is very specific ("In the neighborhoods where they're running businesses or collecting protection money, you won't see people getting mugged because the yakuza don't want people to be afraid to come there and spend money") and doesn't make much sense in the context of things like the Economist's article As crime dries up, Japan’s police hunt for things to do:
THE stake-out lasted a week, but it paid off in the end. The tireless police of Kagoshima, a sleepy city in the far south of the country, watched the unlocked car day and night. It was parked outside a supermarket, and contained a case of malt beer. Finally, a passing middle-aged man decided to help himself. Five policemen instantly pounced, nabbing one of the city’s few remaining law-breakers.
and
Even yakuza gangsters, once a potent criminal force, have been weakened by tougher laws and old age.
This gels with my experience. Aichi police cracked down on Yakuza in the last ten years especially. Having said that, An acquaintance (bar owner) in Shinjuku openly told me last year that he was worried that some day they would show up and demand protection money.
