The title used here is meant to be provocative, but the essential point is that we have historical precedence for what happens when we close our borders for those most desperately in need: people die.

Those who do not remember the past are condemned to repeat it.
--George Santayana

History Buff details the parallels, here:

According to documents discovered in 2012, Anne’s father wrote numerous letters to U.S. officials pleading for permission to immigrate with his family. Most of these were written between April and December of 1941, in the months following the Nazi’s occupation of the Netherlands, where the Frank family had been exiled.

At the end of 1941, after the family’s pleas were ignored, the Franks would go into hiding for two years, until they were eventually discovered and sent to Auschwitz. In 1945, Anne and her sister Margot died of typhus in the Bergen-Belsen concentration camps.
...

Following World War I, Congress passed the Immigration Restriction Act of 1924, which set up a quota system that discouraged the immigration of Jews, Italians and other “undesirables.” For anyone following the GOP primaries, the rhetoric here will ring familiar: Harry Laughlin, superintendent of the Eugenics Record Office, told Congress that new immigrants were polluting the country’s bloodline with “feeblemindedness, insanity, criminality, and dependency.”

Despite the existence of this harsh, racist quota, things managed to get worse in the years leading up to World War II. At the behest of American consular officials and the Assistant Secretary of State, a bureaucratic maze was constructed to ensure that even the meager quota remain unreached. The political “paper wall” was bolstered by the public, as anti-semitism spread throughout the United States along with the fear that the Jews had been infiltrated by Hitler’s spies. Between 1941 and 1945—the height of the Nazi regime’s systematic killing—90% of the available spots were unused, translating to nearly 190,000 lives that went unsaved.

(translated: “This is a photo as I would wish myself to look all the time. Then I might have a chance to go to Hollywood. But I’m sorry to say that I usually look different these days.”
Anne Frank, 18 October 1942)