I believe that if Putin does nothing (publicly) then it's a green light for others to start doing it too.
Robbing Russian Mafia is a task not for the faint of the heart. But that's minor details; the proposal failed to pass anyway. I think Putin now has a good possibility to position Russian banks as a safe haven - safe from EU's turbulence, at least. Russian economy is stable, thanks to high oil and gas prices. Additionally, the same well known team is in power - and they are not likely to do something exceptionally stupid with other people's money. At least because they don't need to. Perhaps Mafia figures won't be so interested to keep their ill-gotten gains in the country; however plenty of foreign (EU) clients might not care that Putin can look into their accounts.
Probably China and other major non-EU players are also interested to get into this game. They are a bit farther geographically, though - however little it matters today, in the age of instant secure communications.