Hi - it's Customs and Border Protection's responsibility. A lot of times the cruise lines are clueless about being in violation...especially if the cruises are booked separately. Then they turn their passenger manifests over a couple of weeks before the cruise. That's when you can usually expect a call that you have to cancel if the cruise line didn't catch it at first.
Nobody orders the cruise line to bill the passenger, it's the ship that is doing the violating and is fined. Ships just pass this along to the passengers.