If I recall correctly, FDR hailed from an affluent family, from the state of New York, with the state of New York being a solidly Republican state, regarding presidential elections, throughout the 19th and 20th centuries (with the exception of its support for Woodrow Wilson in the election of 1916). Moreover, his cousin, president Theodore Roosevelt, was obviously a member of the Republican Party. Thus, his family would have presumably been affiliated with the Republican Party.
I had posed this question to my AP U.S. History teacher several years ago, in high school, and she asserted that FDR was a Democrat because Herbert Hoover had run for re-election in 1932 as the incumbent candidate from the Republican Party. Though it would be prudent for FDR to have run against Hoover for this reason—as FDR would likely have been precluded from garnering the nomination for the Republican Party, due to the establishment’s presumable support for the incumbent candidate—this assertion doesn’t seem veritable, as FDR also served as James Cox’s running mate in the election of 1920, and, therefore, had an enduring affiliation with the Democratic Party. So, why was FDR a member of the Democratic Party?
Well your history teacher is definitely wrong. Roosevelt was running for office as a Democrat 22 years before that election. It's not like he jumped parties. He wasn't even breaking away from his immediate family since while his cousin Theodore was Republican, until he wasn't, Franklin's father, James, was Democrat connected ever since James Buchanan, who was the ambassador to the United Kingdom hired him to work as secretary. Buchanan went on to be the second worst president in American history but the connection was made.
James Roosevelt went on to become a town supervisor for the Democrats, and the only reason he didn't go on to running for higher officer was because he was too busy being rich. He went on be a close ally of another Democratic president, Grover Cleveland and Franklin's brother James Jr got appointed to be head of the United States legation in Vienna.
Long story short, Franklin was a Democrat because he was from a family of Democrats, because his political connections were to the Democratic Party, and because it was the Democratic Party who approached him and said "Hey! Wanna be our candidate for State Assemblyman?"