A natural way in this situation is to make deliveries less often; we no longer need official letters urgently, except in quite rare exception cases that can probably be covered by parcel delivery services, with tracking (and much higher cost for the small number of deliveries).
Over here, the postal service tries to solve this by arranging deliveries only three times a week, and in future perhaps once a week. I find that quite acceptable. If information needs to reach me faster, it'll come on-line anyway. A week's wait is OK for legacy services where someone wants a piece of physical paper with my signature, written on a pen, on it.
There is always a limit on how much money you can get from the government so they would have to save on expenditure with the decline.
It's nice to have but it can't be considered critical anymore. Bills, paychecks and other important mail needs to be delivered digitally, there's no turning back.
edit: parcels and legacy papers still need to work everywhere though and that is worth some public spending, but if you live in the middle of nowhere you have to accept longer deliveries on fewer days of the week.