Why would you call most of pre-modern America "lawless"?

Bank notes had an issuer listed and were (supposed to be) redeemable in coin. So the way to enforce it was to show up at the bank and present the note for redemption. A bank note is essentially a check payable to bearer (negotiable paper that doesn't have to be endorsed to pass to a new holder), so any penalties for a bad check is conceivable for a bad note. I don't know what the exact penalties were, though. Also, there were occasional panics or crises when redemption ended for almost everyone, and of course any particular bank could go belly-up.

Redemption isn't useful if it's says it's from the Second Bank of East Squattachuck, New Hampshire, and you're in Columbia, South Carolina. I know that, at least near the end, there were large books published describing notes of various banks. The verification problem had happened with negotiable paper for centuries, and possible solutions were to refuse the note, take the word of some reputable person that it's good, or accept it at a discount from face value (the discount representing the risk that it's bad). But David R. Johnson estimated wrote that "when the Civil War erupted, perhaps as much as half of the paper notes in circulation were counterfeit." (Presumably from his Illegal Tender : Counterfeiting and the Secret Service in Nineteenth-Century America) This review of Johnson's book says "Paper money consisted of the printed notes of almost 1,400 banks chartered in the individual states. Counterfeiting was rife. According to the New York Times [in 1862] up to eighty per cent of the banknotes in circulation were forged. The profusion of banknote issues and the reliance on local agencies for law-enforcement made counterfeiting quite easy to get away with .... Catching [people distributing counterfeit currency] was, in any case, rarely a priority. People finding faked notes were more likely to palm them off on others than to complain, and governments were largely indifferent to the matter."

All this ended with the National Bank Note system enacted in the US Civil War, the prohibitive tax levied on private bank notes, and the use of the Secret Service to go after counterfeiters.

This book matches something that I remember reading elsewhere, that the Banks of the United States had a policy of immediately redeeming local banknotes, thereby being an indirect regulator, and this annoyed powerful interests.