A little background on me to give this question context. I'm from New England, my dad is white and my mom is black, yesterday I was talking to my grandpa about what it was like for him growing up in a small town in Western Massachusetts in the 1930s, 40s and 50s (I have some questions about that as well, he told me that he didn't know there was prejudice growing up, he had mostly white friends, went to a mostly white church, his family had dinner at white neighbors homes and, in his words he "got teased for looking like a dago or some kind of Portuguese more than for being from thr negro race" and I am curious if this was a common thing for New England at that time, WEB DuBois grew up one town over a few generations previous and also apparently had a very good experience, but that's for another post), and the topic of politics came up. He's a Republican, he's been a Republican his whole life and said the only time he voted Democrat in any election was in 1964's presidential election, I started asking him what were the things he believed that made him a Republican and he said that he was against corruption and for merit, against monopolies, good government and the constitution, that he supported the New Deal but thought that it made the government too large and encoraged corruption and idleness. He said a lot of other things about politics but for this question I think these are most relevant. To me this is very confusing, I thought the Democrats were the New Deal party and the GOP was against the New Deal, but I have been reading that there were many Democrats against the New Deal and Republicans for the New Deal and they formed cross party coalitions against each other. I knew that there was a rift in the Democrats about Civil Rights that caused many kg them to work with the GOP which was wholly pro-integration, but I thought that was the exception.
So, what were the key issues that mace someone choose to be a Democrat or Republican?
First of all--
I knew that there was a rift in the Democrats about Civil Rights that caused many kg them to work with the GOP which was wholly pro-integration, but I thought that was the exception.
Politicians' stance on civil rights were almost solely determined by whether they were from the North or the South. Democrats held almost all of the seats in the South in both the House and Senate, so overall they were more likely to vote against civil rights measures, but northern Democrats were actually the most supportive group.
For the Civil Rights Act of 1964, around 95% of northern Democrats, 85% of northern Republicans, 10% of southern Democrats and 0% of southern Republicans (a tiny group, admittedly) voted yes.
Now for the main question-- then as now, a large part of what determined your political party and your opinions on the issues was your socio-economic background.
First, white Southerners monolithically backed the Democratic Party due to the legacy of the Confederacy and to ensure there would be a strong national group to push back against integration.
African-Americans had, coming out of the Civil War, mostly voted for Republicans where they were allowed to vote, but the New Deal mostly brought them around to the Democratic Party, although black Republicans still existed in significant numbers until the passage of the Civil Rights act.
Members of the working class also primarily voted for Democrats, for several reasons-- they mostly supported the New Deal, and Democrats were strongly associated with the union movement, which had a lot more sway over the votes of their members than it does now. In many cities, Democrats also had strong political machines, often affiliated with white ethnic and usually Catholic populations (Italians, Irish, Polish, etc.).
On the flipside, Republicans had strong support among wealthier and more educated people, including in cities, and very strong support in suburbs and protestant rural areas-- some of that was a heritage of the civil war, some of it came out of the big political dividing issues of the early 20th century, immigration (which was supported by ethnic Whites in the cities and opposed by most of the more rural communities) and prohibition (where Democrats were mostly "wets", advocating for an end to prohibition, and Republicans were "drys", popular with Protestants who mostly had strong values against alcoholism).
Another big dividing issue going into WW2 (so, covering the earlier part of your timeline) was that Republicans were mostly isolationist, advocating against an intervention in Europe. The electoral effects of that were complex, but some of the groups that tended to appeal to were German-Americans, traditionally a swing voting group, and women, who at that time were actually generally more Republican-leaning than men (also because they were less directly involved in labor unions and because they tended to strongly favor prohibition).
There's obviously more than that, in many states local issues tended to play a role as well, but that's some of the broad strokes. It's definitely true that the parties were not as ideologically homogenous as they are now-- there were liberal and conservative Democrats and Republicans, and many political commentators of the time already foresaw a future realignment into a "Liberal Party" and a "Conservative Party", so your choice of party was in some ways even more influenced by identity politics than it is today.