Most do.

For example: https://www.reddit.com/r/Banking/comments/1csl00q/any_banks_...

4.2% of US _households_ are unbanked. Banks refuse customers for many less extreme reasons than “can’t verify identity” which is in fact a requirement for them with all of their federal and state regulators.

>How many billions would a major bank loose in a lawsuit, if they did the sensible thing and refused to deal with 3% of the population?

Zero. Because FINCEN/KYC[0] laws in the US mandate identification for all customers.

Which means that at least 3% of the populations is "unbanked"[1]:

   Some reasons a person might not have a bank account 
   include:

   Lack of access via a nearby bank branch or mobile phone

   Minimum balance fees

   Distrust of the banking system, typically due to lack of transparency    
   regarding fees and deposit timing[1]

   No access to government-issued ID, which is required to open a bank account

   To avoid delinquent debts, such as creditors seizing the account in 
   judgements, or the government collecting back taxes or child support

[0] https://www.investopedia.com/terms/k/knowyourclient.asp

[1] https://en.wikipedia.org/wiki/Unbanked

Edit: Fixed subject/verb agreement (laws/mandate)/fixed quote formatting.

Further reference for the ID requirements:

https://www.bankhelp.gov/help-topics/bank-accounts/required-...