If a teller was suspicious, they would compare your signature with the signature they had on file. Later on, you would also usually need to know the account holder's address in order to complete a withdrawal slip or a check.

Banks would maintain large books of their account holders' signatures or, if the account holder was illiterate, their "mark". Some banks would even have a combination, where they would record the account holder's signature with an attached "secret mark known only to the writer and the bank." (Source: "Old Books Worth Fortunes", The Decatur Herald, May 14, 1904.)

Bank forgery became a real problem in the late 1800s and early 1900s. A March 13, 1913, article in The Chicago Inter Ocean newspaper called bank forgery "the easiest and most profitable crime of modern times." This is one of the reasons that photo IDs began to be required.

Further reading:

"Bank Depositors Compelled to Look Out for Forged Checks", The New York Sun, Aug 31, 1913

"Science of Crime", The Atlanta Constitution, Feb 26, 1893

"Postal Savings Bank Forgery Is Charged", Nevada State Journal, Jan 14, 1914

"Police Get Suspect In Big Forgery Case", Great Falls Tribune, Apr 8, 1918

Something not mentioned thus far, is the passbook. A small ledger of a customers deposits and withdrawals. That was a standard banking object from the 18th c until about 20 years ago when they seem to have nearly vanished all together.
There was only one copy and it was held by the customer, and presented to the bank when making transactions.
Unless something else was suspect, the mere possession of this ledger was considered proof that you were authorized to make transactions on that account. Losing your passbook was a big headache.
I think I still have one from an old savings acct in my desk drawer.