LOS ANGELES — With no end in sight to the federal government shutdown, governors across the nation are struggling with a cascade of tough decisions about when and whether to step in with state money to keep shuttered parks and programs operating until the deadlock is resolved.

Here in California, Gov. Jerry Brown, a Democrat, said the state, which is recovering from a steep economic decline, would not intervene to keep its vast network of national parks, like Yosemite, open to the public until the federal shutdown ended.

But in South Dakota, Gov. Dennis Daugaard, a Republican, has pleaded, so far unsuccessfully, with the Interior Department for permission to use state money to keep open the Mount Rushmore National Memorial, a symbol of his state and one of its top tourist draws.

“Mount Rushmore is just a wonderful tourist destination and clearly the most significant attraction for tourism in South Dakota,” Mr. Daugaard said in an interview.