The 10-year-old was the subject of some mocking on social media during Trump's inauguration. One "Saturday Night Live" writer was suspended from her job over an insulting tweet she posted about Barron.
"It is a longstanding tradition that the children of Presidents are afforded the opportunity to grow up outside of the political spotlight," the statement said. "The White House fully expects this tradition to continue. We appreciate your cooperation in this matter."
The focus on Barron brought derision from both sides of the aisle.
Chelsea Clinton, daughter of former President Bill Clinton and former Democratic presidential nominee Hillary Clinton, posted a statement on her Facebook page over the weekend encouraging people to lay off Barron.
"Barron Trump deserves the chance every child does - to be a kid," she wrote.
Barron is the child Trump has with his wife Melania. He has other grown children from previous marriages.