Silver is money for catastrophic times, when banks are padlocked and the entire electronic transfer of funds becomes an impossibility, paper money will be a chancy proposition at best, and probably refused in many transactions.
There shall always be barter, as well. But silver and gold, they ring true in the estimations of many, enough so that these metals will be coveted in desperate times. If not, they fashion into some rather nice looking jewelry and display pieces.
Silver doesn’t seem to play much of a role on the Walking Dead. You can’t eat silver!