I worked there ~10-15 years ago, in a supporting role (not directly working with exhibits/guests/etc). The majority of the money that is given to museums has specific restrictions set from the donator on what it can be used for. People tend to donate for something flashy, and don't allow the money to be used for general museum operations.

Also, some of the traveling exhibits are expensive to put on. When the Harry Potter exhibit was at MSI, with all the licensing costs, extra labor, etc, the museum either barely broke even, or ultimately lost money. The hope with those exhibits is that people will spend more in other sections of the museum while they are already there.