Since nobody carries cash and most businesses (begrudgingly) have a small amount of cash, robberies, pick-pocketing, etc. have basically vanished as an entire category of crime.
(Other crimes are also on their way out, but for far more troublesome reasons.)
That being said, your parallel doesn't really work. You want to keep cash because your society is broken. The crime rate dropping wasn't the goal of going cashless, it was a side effect.