The Republican governor, in a letter to the National Park Service, said state employees would provide security at the monument. He would lead a drive to raise private funds for lighting, the governor's office said in a news release.
Under the plan, the visitor's center would close but concession stands and the gift shop would be open.
"Mount Rushmore is a national treasure and is a special location for South Dakotans and citizens across the country," Daugaard said. "I am confident state government could work with the federal government and private vendors to securely and competently manage Mount Rushmore during a federal shutdown."
If there is no agreement in Congress on funding the federal government when the new fiscal year begins Tuesday, operations considered non-essential would close. The closing of national parks and monuments would be one of the most visible signs of the shutdown.
Mount Rushmore in the Black Hills, featuring the heads of Presidents George Washington, Thomas Jefferson, Abraham Lincoln and Theodore Roosevelt carved from a cliff, attracted almost 3 million visitors in 2012 even though it is remote from major cities.
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