Most banks require you to enter a SSN or another TIN to enable their online services. This makes it almost impossible for persons that have no SSN or ITIN to use online banking.
As a remedy of this problem, it is usually also possible to enable online banking by calling designated phone numbers. There you will be asked personal information, that is on the bank's record to identify yourself and have the online banking enabled this way.
It should be obvious that online banking may also be enabled while opening the bank account at your branch. But most astonishingly not all banks offer this.