What is your purpose for wanting a Swiss bank account? If you want one because you believe it is more private then a US bank account then what you want is known as a numbered account. Information concerning such accounts is restricted to senior bank officers, rather than being accessible to all the employees of a bank. However, the information required to open such an account is no different from that of an ordinary account; completely anonymous accounts are prohibited by Swiss law. Should a criminal investigation take place, law enforcement has access to information related to a numbered account in the same way it has access to information about any other account. Also a Swiss bank or any other business isn’t going to go through the trouble of setting you up with a numbered account if you’ve only got a couple thousand to deposit. If you’re looking for something along these lines then I would suggest checking out
this site. If however, you’re just looking for a good Swiss bank to open an account with then both UBS and Credit Suisse are excellent Swiss banks that operate in the US and have many local branches.