The demonstration was organized by Native American protesters, according to Lee Strubinger of South Dakota Public Radio.
In a video shared with Reuters, several white vans blocked the road and protesters criticized Trump's visit for increasing the risk of spreading the COVID-19 virus and for celebrating US independence in an area that is sacred to them.
Mount Rushmore, which depicts United States presidents George Washington, Thomas Jefferson, Theodore Roosevelt and Abraham Lincoln, has not hosted a fireworks spectacle since 2009 because of environmental concerns.
Trump Jr. tested negative, according to the report, and the couple has not been in contact with the commander in chief.
"Make no mistake: This is being done deliberately to discredit America's founding principles by discrediting the individuals who formed them", she said.
"I stand in solidarity with the Lakota Sioux & this is NOT ok with me", he added. It did not overwhelm emergency services as many had feared, but one senior police officer said Sunday it was "crystal clear" that drunk people struggled, or ignored, social distancing rules.
"I want people to stop calling us racists", Lee said.
The Trump administration said it would be the biggest display in living memory.
In his high-profile Independence Day speech, President Donald Trump warned against the dangers posed by BLM protesters, frequently dubbed "left-wing fascists", in light of the recent destruction on the country's streets and the way numerous demonstrators approach United States history. The virus causes COVID-19, a disease that primarily requires hospital care in the elderly and people with compromised immune systems.
Trump is giving a pre-Fourth of July speech at Mount Rushmore in South Dakota.
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His comments came after a sudden, recent shift in recent days in which Republican lawmakers and governors have begun strongly encouraging mask wearing - including urging the president to don one to set an example for Americans. Event organizers were to provide masks to anyone who wanted them and planned to screen attendees for symptoms of COVID-19.
Mr Trump has predicted a "fireworks display like few people have seen" in South Dakota, a state he won in the 2016 USA election.
The President was addressing thousands of his supporters from beneath Mount Rushmore on the eve of America's Independence Day and he claimed that left-wing protesters were trying to erase American history and overthrow the American Revolution.
"We're going to stay with each other", she said.
"The president is putting our tribal members at risk to stage a photo op at one of our most sacred sites", said Harold Frazier, chairman of the Cheyenne River Sioux Tribe.
More than 100 protesters, many Lakota, lined the road leading from Keystone to the monument holding signs and playing Lakota music in 95-degree heat. "It's an injustice to actively steal Indigenous people's land then carve the white faces of the conquerors who committed genocide".
"The president needs to open his eyes".
Actress Mia Farrow has also repeated her disgust with Trump.
The former Fox News host tested positive for coronavirus after attending two Trump rallies in Tulsa, Oklahoma, and Phoenix, Arizona, both of which had no social distancing or mask requirements.
"American heroes defeated the Nazis, dethroned the fascists, toppled the communists. and chased down the terrorists to the very ends of the Earth", Trump said at the White House in his speech. While the president had previously expressed his intention to hold a large-scale event on the National Mall with military displays as he did last year, the White House said this year's celebration was modified because of coronavirus concerns.