It's quite obvious that when you decriminalize something, the crime rate will go down, since the people doing that thing are not criminals any more. (n / m) < (n - 1) / m.
But even apart from that (because it's a cooking-the-books level trick, really), decriminalization of drug use has other effects that lower crime rates. For example: keeping up a heroin habit is hard when you need to pay black market rates (€100 / $150 per day). So often, these people need to resort to burglary, robberies or prostitution. Regulating the market drives the prices way down (you could have retail prices for a brick of coke that are the same as that of a package of coffee if you'd let the market work unrestrained), which causes people to not have to resort to crime to fund their habits.
(note: not advocating unregulated heroin or cocaine markets, just stating the obvious socio-economic implications of prices within the bounds of affordability for regular people with a regular job, leaving aside their ability to do said job)
But even apart from that (because it's a cooking-the-books level trick, really), decriminalization of drug use has other effects that lower crime rates. For example: keeping up a heroin habit is hard when you need to pay black market rates (€100 / $150 per day). So often, these people need to resort to burglary, robberies or prostitution. Regulating the market drives the prices way down (you could have retail prices for a brick of coke that are the same as that of a package of coffee if you'd let the market work unrestrained), which causes people to not have to resort to crime to fund their habits.
(note: not advocating unregulated heroin or cocaine markets, just stating the obvious socio-economic implications of prices within the bounds of affordability for regular people with a regular job, leaving aside their ability to do said job)