Anyone in the museum profession will tell you that it's expensive to keep a museum running. Just keeping the power on can be costly to smaller institutions. Preservation, storage and management of collections, at the heart of many art and history museums, can be a challenge to fund. Then there's also educational programs, public programs, group tours, etc, that all cost something to provide. And there's always something else we wish we could do, if we had the money.
The public seems to mainly object to two things: staff salaries and high admission prices at institutions that receive or have received public funding. Neither objection is unreasonable, though much of the public does not know where museum money goes because most of the information is usually not available. Perhaps if a budget displaying where the money goes was shown next to how much it costs to get in, the public would at least have the opportunity to understand how the money is used.