A Social Security number is a federal identification number for tax purposes. It is not a work permit. If you are an international student holding F-1 or J-1 status, the Social Security Administration (SSA) will only issue a Social Security card to you if you have been offered an on-campus job or have been authorized off-campus employment.
It is not necessary to have a Social Security number to open a bank account. The SSA will not issue you a Social Security card for banking purposes.
The Department of Motor Vehicles (DMV) normally requires a Social Security number to obtain a California Driver's License; however, international students are not eligible to obtain a number for this purpose. If the DMV tells you that a Social Security number is required, you must tell them you are not eligible for a number. They should then allow you to proceed with your application.