Both were in a bad place, but the GOP was objectively worse off in the 1930s than the Democrats were in the late 1860s. Consider:
1868Rep (Grant): 52.7% Dem (Seymour): 47.3% Margin: ~300,000 Votesversus
1936Rep (Landon): 36.5% Dem (Roosevelt): 60.8% Margin: ~11,000,000 VotesThe pv margins are what's important here: considering the baggage weighing down the Democratic party (the stigma of having opposed the Civil War + Andrew Johnson's disastrous tenure), Seymour ran a very strong campaign and came close to defeating Grant, the biggest war hero since Andrew Jackson. By contrast, Landon barely even registered against FDR, loosing by close to 11 million votes. Even accounting for the massive population growth between 1868 and 1936, there's hardly a comparison. Had anyone but Grant been the nominee in 1868, Seymour might very well have won.
The reason for this is that, during Reconstruction, the Democrats were able to remain competitive by appealing to racist fears of "miscegenation" that made many Northerners hesitant to vote Republican. That they had opposed the war made them distasteful, but people ultimately vote their own interests . It was much easier for Democrats to convince Northern whites that it was in their interest to keep freed slaves from coming North and taking their jobs than it was for Republicans to convince Americans that it was in their interest to vote away the New Deal.
EDIT: Also to be considered is how long it took for each of these parties to regain dominance after their initial defeats. Samuel Tilden, the first Democrat to win the popular vote in the post-Civil War era, did so in 1876, 16 years after Abraham Lincoln was elected president. By contrast, the first Republican to win the popular vote after FDR's first victory - Dwight Eisenhower - did so in 1952, 20 years later. That Eisenhower was a war hero who probably would have won no matter which party he ran for, while Tilden was a relative newcomer to the national stage, further supports this.
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