How much should it cost to go to a museum?

A recent lawsuit against the Metropolitan Museum of Art challenges the museum on its admissions policy, and other museums are taking notice. BBC Travel notes that nearly half of the US art museums it surveyed don’t charge admission, which they note tends to correlate with a city not being considered a tourist destination (i.e., the Cleveland Museum of Art doesn’t charge admission, but the Boston Museum of Fine Arts does.) Should museums charge admission?