You have it mostly right, the Democrats and Unions become tied together during the Great Depression. There are several major reasons for this. First, few major politicians actively supported labor unions as party policy. Progressives in the early 1900s passed several laws that protected the workers rights. Things like Workmen's Compensation laws are passed at the State level by 1913 (22 States), Keating-Owen Child Labor Act (1916) is a federal act outlawing child labor, Adamson Act (1916) regulates an 8 hour work day for the railroad industry and most importantly is the Clayton Anti-Trust Act of 1914 which states "that the labor of a human being is not a commodity or article of commerce, and permit[ting] labor organizations to carry out their legitimate objective." (https://catalog.archives.gov/id/6171421).

But all of those pieces of legislation simply nudged the government towards an even playing field when it came to labor-business relations. Remember, it was in Gilded Age just, in some cases, 20 years prior to the passing of those laws that the government almost always came down on the side of business going as far as calling out state militia/national guard to suppress strikes (Homestead Strike as an example - https://www.aflcio.org/About/Our-History/Key-Events-in-Labor-History/1892-Homestead-Strike). Even these seemingly pro-union or at least pro-working class pieces of legislation were mostly overturned by the Republican government of Harding in the 1920s.

What occurs in the Great Depression, besides the obvious millions of Americans suddenly unemployed, is a shift in how the federal government saw its role in terms of populace-government responsibilities. And really it is more than just a governmental shift, it is the birth of the truly modern labor unions as well.

Government wise, FDR realized the push for socialism was creeping across all of the Western democracies. It was the new thing, and apparently it worked...or so the average American might have thought at the time. Thousands of Americans applied to work in the Soviet Union because that is where the jobs were. So for him, it was a tightrope walk allowing some elements of socialism while retaining the US commitments to republicanism. While many others (Huey Long, Upton Sinclair, Charles Coughlin, Francis Townsend, etc.) are calling for a greater amount of socialism, FDR paints his plans as for the working class by directly employing millions of Americans while still keeping true socialism and fascism at bay. Organizations like the CCC and WPA are the federal government's first attempt to employ mass amounts of Americans and so Roosevelt (and thus the Democrats) are seen as a friend of the working-class people. In addition the passage of the Wagner Act which establishes the National Labor Relations Board and the Fair Labor Standards (1938 - Bans Child produced goods, sets an hourly minimum wage, and requires overtime for anything over 40 hours/week) solidifies this belief of Democrats as the champions of labor.

Labor Union wise, the CIO (Congress of Industrial Organizations) was created in 1937 and some high profile strikes like the "Battle of the Overpass" in Dearborn, Michigan and the sit-down strikes in Flint, Michigan draw national attention to the plight of unions and their members. GM begrudgingly agrees to give pay raises tied to the standard of living increases. US Steel recognizes United Steelworkers of America before they strike against the company and all in all by 1940 Union membership reaches 9 million. The CIO really becomes one of the major driving forces in labor union politics.

So to answer you succinctly, the Democratic Party attempted to tie themselves to unions midway through FDR's presidency (1936-1938) as part of the second New Deal. Other parties, political ideologies, and individual politicians prior to the Great Depression could be seen as "friends of labor" but very few thought it was the government's responsibility to completely side with unions.