A refugee isn't just anyone who wants to immigrate because they long for the bucolic splendor of Minnesota in mid-February. They've been forced to leave their home due to war, persecution, etc., and they have legal rights under international law. So if a bunch show up at your border, you can't shove them in a shipping container and throw a few loaves of bread in every week. Conversely, countries taking in refugees can legally subject them to security screenings (though Fox News wants you to think you'll be forced to take an al-Shabaab militant as a roommate). While criminals do try to manipulate the process, attempts are rare. America took in 800,000 refugees between 2001 and 2016, and later arrested a grand total of five of them on terrorism charges. Worldwide, refugees are far more likely to be the victims of violence than the perpetrators, because their vulnerable status makes them easy to scapegoat.
But that scapegoating can sure be convenient if your disdain for helping foreigners is a feature and not a flaw for your supporters. In 1942, a ship full of Jewish refugees arrived in New York City, and a Gestapo agent was among them. He was caught during the screening process and put on trial, and that was all that was needed to start the scaremongering. An American ambassador blamed the fall of France on a vast refugee spy ring, and Roosevelt himself argued that refugees had been coerced into spying -- all without a shred of proof.
In 1938, America received 125,000 visa applications from Germany and annexed Austria, but only accepted 27,000. And restrictions only tightened as the war went on and the immigration crisis grew. By the end of 1944, at which point the Holocaust was becoming well-known, Roosevelt was advised to reject all refugees. A Treasury Department report that called the State Department "guilty not only of gross procrastination and wilful failure to act, but even of wilful attempts to prevent action from being taken to rescue Jews from Hitler" finally convinced Roosevelt to accept tens of thousands of people. But that wasn't much help to those who had already been rejected and forced back home to face death.