At least here in the US, the FBI would get involved and depending on how much damage and chaos it causes would determine the manpower they assign to that case.
Banks themselves would either make the state add extra security to each bank or hire their own security.
Insurance claims would be filed if they can be filed, these would help the bank cover the damage costs.
Security would keep getting escalated if it kept happening, eventually getting to an active manhunt for the culprit and heavily armed guards at banks.
Any sort of data wouldn't be lost due to offsite backups (This is if they even have a half-decent IT staff).
This is what I believe would happen.