Before Theodore Roosevelt, no President of the United States left the country while in office:
"Prior to President and Mrs. Theodore Roosevelt's visit to Panama in 1906, no American president had set foot outside the country during his tenure in office, not even crossing a bridge to Canada or Mexico. In an August 1906 letter to Andrew Carnegie, President Roosevelt bemoaned the 'ironclad custom which forbids a President ever [going] abroad' that kept him from engaging in direct personal diplomacy with 'the Kaiser and the responsible authorities of France and England' to negotiate events of the day.1 Considering Roosevelt's view of the United States' waxing role in global influence, his desire to witness construction of the Panama Canal--something he considered his 'most important [presidential] action . . . in foreign affairs'--is perfectly understandable.2
"There were multiple reasons for a president to remain inside the nation's continental borders. The historian Richard J. Ellis argues that while technology limited travel options for the earliest presidents due to both pace of travel and lack of communications, presidents were equally 'constrained by norms and expectations that were rooted in the nation's understanding of its role in the world.'3 Americans eschewed 'entangling aliances' (George Washington's term) [actually, it was Jefferson's--DT] with European nations, beyond trade agreements, and abhorred 'European monarchy and autocracy.' The nation believed that the unique U.S. system saved it from 'the path of war, conquest, taxation, and empire.' According to Ellis, 'A republican president was to exhibit simplicity of manners; he was to be the nation's 'first citizen'--distinguished and admired but a fellow citizen nonetheless.'4 Thus Americans had historic fears that, combined with the travel and communication complications, kept presidents home and limited their contacts with European royalty and its potentially negative influences. Roosevelt clearly recognized the national tenets that constrained him but was determined to visit nonetheless...
"The trick was getting around the 'ironclad custom.' The administration of President William McKinley helped make that possible. McKinley not only expressed interest in going outside the national boundaries, but the imperialist United States that matured during his terms in office--mostly owing to the 1898 war with Spain--helped establish the view that 'imperialism [was] no longer . . . a perversion of America's mission but its fulfillment.'7 An assassin's bullet cut short McKinley's plan to visit Hawaii, Puerto Rico, and Cuba during his second term--all American territories or lands under U.S. protection--or he might have been the first president to travel abroad instead of that distinction going to the vice president who succeeded him...".
https://www.whitehousehistory.org/off-for-the-ditch
Question: What if the "ironclad custom" had been enshrined in the Constitution? Probably the Framers would not put in an absolute prohibition, recognizing that sometimes such trips might be advantageous or even necessary--but perhaps they could require such trips to have congressional approval, maybe even a supermajority, perhaps even a two-thirds vote of the Senate. This could certainly have consequences in 1919, when many people in Congress (and elsewhere) did not like the idea of Wilson going to Paris...