You don't need a SSN to open a bank account. But if you receive interest payments reportable on a 1099 and don't give the bank your SSN the payments can be subject to backup withholding.
From the IRS website:
When you open a new account, make an investment, or begin to receive payments reportable on Form 1099, you must provide your TIN. For certain types of payments, you must provide the TIN in writing and certify under penalties of perjury that it's correct. In those cases, the bank or business will give you Form W-9, Request for Taxpayer Identification Number and Certification or a similar form. If your account or investment will earn interest or dividends, you must also certify that you're not subject to backup withholding due to previous underreporting of interest or dividends.
You may be subject to backup withholding and the payer must withhold at a flat 24% rate when:
you don't give the payer your TIN in the required manner