According to this page from the Holocaust Museum,
Sailing so close to Florida that they could see the lights of Miami, some passengers on the St. Louis cabled President Franklin D. Roosevelt asking for refuge. Roosevelt never responded. The State Department and the White House had decided not to take extraordinary measures to permit the refugees to enter the United States. A State Department telegram sent to a passenger stated that the passengers must "await their turns on the waiting list and qualify for and obtain immigration visas before they may be admissible into the United States." US diplomats in Havana intervened once more with the Cuban government to admit the passengers on a "humanitarian" basis, but without success.
FDR's action was supposedly based on the 1924 law.
But did he have the power to override the law on humanitarian grounds, and did he choose to forgo its exercise? It appears he did. In August 1940, the Portugese freighter Quanza, a ship filled with over 300 passengers, including 83 Jewish refugees fleeing from France, was allowed to dock in Norfolk, Virginia. Asylum was granted to all passengers — many of whom at the time believed they had to return to Europe, and the Nazis. Eleanor probably deserves some credit for the policy change.
The man who became president for life, was hardly known as timid with his presidential power. If he rejected them, then he rejected them on purpose.