There are certainly some private banks that are not very choosy where they take their money from. There are more than a few instances where payments have been made to corrupt people from within highly respectable banks from those wanting to park money they otherwise would not be able to place in a respectable banking institution.
Private bankers turn down a lot of dubious customers. A client's secrecy is fine up to a point. Naturally we respect privacy, but some customers are looking for confidentiality. We are bound to disclosure under certain conditions and laws. We pride ourselves on imposing rigorous standards on our staff to make sure they avoid suspicious transactions. If somebody shows up with $1 million in cash and the circumstances are unclear, then we won't accept it. We cannot afford to have a scandal. We would far prefer to turn down money than run the risk of impugning our reputation. Our staff are also well aware that failure to comply with the law could result in substantial criminal penalties, including jail time.
Private banks are more used by individuals who have received 'grey' money through kickbacks and other forms of corruption, rather than large-scale money launderers (who would instead use remittance-houses and set up their own financial companies to transfer and reinvest dirty money).