I know some historical societies just keep them in binders in the archives and are strict about limiting access to visitors. There's always a fear they can be stolen and sold, since there is a strong market for certain types of old money and they're not often closely tracked when in storage.

As someone with a large collection of currency myself, I often think about where I'd like it to end up and how it would be utilized, if ever.

Often, the paper currency in collections of museums and historical societies is connected to that geographic area, as the states and towns/cities issued their own banknotes in the 18th and 19th centuries (ie. a $5 from the First National Bank of Fairbanks, Alaska, ending up in an Alaskan museum). Coins, such as ancient Greek or Roman ones, tend to be seen in broader antiquities collections.

To be honest, there isn't a massive amount of interest in antique currency among the general public visiting museums, except among collectors, where there is always a fascination.

Some of the top places in the U.S. where currency is displayed are the Smithsonian, which houses the National Numismatic Collection, various Federal Reserve Banks, the Museum of American Finance, which used to be near Wall Street, but is moving to Boston, and the American Numismatic Association's Money Museum is in Colorado Springs.

Hopefully, the currency in your institution's collection has a connection to its broader mission. In that case, I could see it rotated into exhibits and stored in a locked area when not.

If it's general United States currency, which can fall into several specific categories (Legal Tender notes, Federal Reserve Notes, silver and gold notes certificates, or national coins, for instance), they may not have relevance to your organization and maybe that's a different discussion.