This happened to my wife in Japan when we were trying to board the second try of our flight cancelled the day before. I told them "we don't have any Japanese money, and anyways, this was Delta's fault, so just get them to pay." We sat around for a half hour while they got Delta to pay her fine.
Do they not have ATMs at the airport? Seems unnecessarily risky to not just take the opportunity to pay the fine as quickly as possible - visa overstays can easily end in detention in many (especially Asian) countries and not just in the US.
And while the delay might have been the airline's fault and possibly you could have a civil claim against them for damages incurred it is still your responsibility to have a valid Visa and not overstay it and also your responsibility to pay any fines for it (which you may or may not be able to get reimbursed).
They should have taken care of this when they brought us back into Japan after the flight was canceled (they didn’t catch it when the re admitted her into Japan and the visa was invalidated on her way out since it was single entry, and the flight was to the states so most passengers were visa waiver). And anyways, they were so apologetic that I didn’t really doubt that the Delta Japanese ground crew would just take care of it.
The really sad story is that the flight was canceled a second time after that drama, and we were really feeling a lot of emotions over that debacle. Suffice it to say we’ve sworn off delta for awhile.
