Racists were a big part of the Democratic party until LBJ's Civil Rights legislation and Nixon's Southern Strategy began to siphon them off into the Republican party. Between FDR, when liberals started flooding into the Democrats, and Nixon when racists left, the southern conservative racist Democrats were known as "Dixiecrats". Richard Byrd was the last living Dixiecrat left in federal office. Before him, there was also Zell Miller who was a Democrat, but spoke at Bush's nominating convention in the 2004 election, claiming the Democratic party had left him (and they did when his fellow Dixiecrats became Republicans).
Between there, the Democrats were powerful because they housed both social liberals and also racists, which meant they had a significant majority. The only Republican elected President in that stretch was the liberal war hero throw back to Teddy Roosevelt, Dwight D. Eisenhower, and Democrats generally controlled Congress as well. By the 1960's, the liberals took over enough of the party that the Dixiecrats essentially lost power, which is why Nixon was able to get them to switch.
FDR was DEFINITELY racist. At that time, it was unusual not to be racist if you were white. He did both great and terrible things. Given the context of history, his racist actions, while terrible (especially the Japanese internment camps) had less long-term historical impact than the great things he did. As for Truman, I don't know much about his history before he was President. He isn't among my favorite Presidents, though he certainly was an interesting figure. I do know, however, that he reversed FDR's policy and integrated the US military. Regardless of what he had done before, he was probably the least racist President we had had up until that point in history (even Lincoln who opposed slavery, but until Frederick Douglas changed his mind, wanted to send blacks back to Africa).