Do banks do a back ground check on new account holders?

Usually banks would require an identity proof and an address proof while opening an account. Also they may ask for an "introducer" - a person who has an account with the bank already and who can vouch for the fact that you aren't involved in any illegal activities. In all these cases a background check does not happen.

But, after a few weeks/months/years of using the account, the bank suspects malicious or fraudulent or illegal activities in your account, they can conduct a background check on you and even freeze your account and intimate the law enforcement authorities if their check reveals anything illegal.