The former executives at the Zurich branch of Russia's Gazprombank were given seven-month suspended sentences for helping musician Sergei Roldugin, nicknamed "Putin's wallet".
Mr Roldugin reportedly paid in around $30m (£24m) between 2014 and 2016.
He gave no credible explanation of where the money had come from.
Under Swiss law, banks are required to reject or close accounts if they have doubts about the account holder or the source of the money. Mr Roldugin, a cellist, is godfather to President Putin's eldest daughter, Maria.
Three of the convicted bankers are Russian and one is Swiss.
The men, who cannot be identified under Swiss reporting restrictions, said they would appeal against the Swiss court's decision.
The judge in the Zurich court said it was beyond doubt that Mr Roldugin was not the true owner of the money that he deposited.
The four bankers who opened the accounts should have asked questions: Mr Roldugin had no apparent income, so where did the money come from?
They failed to do this - and have been found guilty of violating Switzerland's due diligence laws.