All responsible banks have KYC (know your customer) rules.
They aren't just going to let some company come in and open an account without knowing anything about the people who have signatory authority over the account and funds as well as the source of the funds.
So criminals don't really use banks and anonymous accounts the way you think they do. What they do is operate semi-profitable businesses to launder the money via semi-boring companies that are small enough to not raise suspicion but profitable enough to conceal the sources of funds.
THOSE business accounts are then used to purchase "services and goods" from offshore organizations with their own accounts that are owned by alternative personas.
The idea of a truly anonymous bank account is a myth.
How is it "responsible" to give in to government regulations designed to invade everyone's financial privacy? All banks will do it - ignoring KYC laws is a good way to get shut down - but it's a disgusting thing.
The issue that was faced is that banks could rightfully claim that they had no duty to verify identity and/or the legal status of funds they accepted on deposit. This led to all sorts of shenanigans like fraud, money laundering, and outright theft.
the KYC regulations are a compromise between having the government know EVERYTHING about your financial life. With KYC, it's the bank's duty to know at least a few things about you and your money to make it harder to use the financial system to conduct criminal activity.
I know a lot of folks like to believe in the libertarian paradise where no government can ever exercise power over you and each man is a sovereign unto themselves but the truth is that a true libertarian community wouldn't be someplace anyone wants to live.
The solution is to make prosecutors do their job instead of giving them an easy out, violating privacy to boot. Prosecutors could subpoena the bank and access the account's records, then continue chasing the criminals onwards.
SOOO....continue the madness that existed before we had compulsory KYC rules. The same rules that allowed the financial system to enable the exact criminal activity we'd like to prevent?