There are many reasons to open a bank account with no ID. If you strive to protect your privacy or have money in a tax shelter, you have good reason to not link your money with your identity. Because such bank accounts are not easily traced, the banks that offer them charge customers high fees.
Ask a mainstream banking company if they know who in their business provides numbered bank accounts. These banks tend to have looser ID requirements and tend to protect your privacy much more than ordinary banks.
Apply via the Internet for a numbered bank account. You will be given a number and a process to follow to deposit your cash. Each bank has its own way of transferring funds, so processes and fees will vary.
Log in and do your banking online. Banks that provide numbered bank accounts are not located on every street corner, so you will need to perform most transactions online and with wire transfers.
Procure a banking passport. Companies that act as an intermediary for a government other than your own can send you a banking passport, which in truth is a second citizenship in foreign country. You'll need to send a picture with a name and the appropriate fee, usually a hefty one. Check your government's laws concerning dual citizenship.
Prepare to pay a substantial fee to open a numbered account with no ID. The fee is a charge for keeping your account private and performing a rare service.