Depends on how far into the history you want to go.
Before electronic banking: telephones and meticulous records. There are actually some goofy outfits the ledger handlers used to wear to prevent themselves getting inked. A lot of people were employed just to make sure this system worked and was not abused (by unauthorised abusers at least)
Before telephones: you could just deposit and withdraw in your local bank which kept the records.
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