As with many non profits and charities, if you did a detailed look at the revenue of many museums and performing arts centers (Kennedy Center in WA DC for instance), their funding sources are diverse and various.

But there's a few noteworthy ones like the Smithsonian where I would take a wild guess that the federal government might be 65% or more of their operating budget on an annual basis, giving the government a very large lever of influence.

For those which are a 501(c)3 you can go find their annual IRS tax filings and make an attempt at breaking down their different incoming revenue streams.