If you want to do it properly, it's not just about opening an account offshore. You will want to set up a trust or corporation in the country, as well as making sure that the country is not likely to share corporate records easily. If you just set up a bank account in the Bahamas in your own name, that info would go straight to the US government.
When I was doing research, the best method I've found of setting up offshore banking was to set up a trust or numbered account with a nominee, usually a lawyer, with me being the beneficiary of the funds in the account. This way there is next for layer of security between you and the money, (the lawyer), and while if the government really wants to find who owns the account they can, but it's much more difficult and they will usually only go to the trouble if there is suspicion that the money is being used for money laundering, terrorism, and other serious offenses.