Chase has had notices in branch for weeks that they will require ID for cash payments to accounts (credit cards, mortgages, etc) starting in March or April. I don't remember.
I have yet to receive a letter and I have made money order payments (never cash) to both my credit card and mortgage.
IMO as long as you're only making payments to your own accounts, I don't really see the impact. It's probably as simple as the teller having to tick a box saying they verified photo ID for the transaction.