You might be interested in some threads listed in the FAQ under the section "Changing roles of Republicans and Democrats":

• Can someone address a brief history of Democrats vs Republicans, specifically the change in Dems from the early 1900s being against civil rights to a more progressive party in the 50/60s leading much social change in the U.S. - 66 comments, over 2 years old.
• This thread mostly goes over the party changes between the 1870s and the 1960s.
• It seems that during the Civil War and reconstruction era, Republicans were liberal and Democrats were conservative. When and why did this change? - 17 comments, over 2 years old.
• One of the commenters gives a very short summary about how muddled both parties were historically when analyzed through through a conservative/liberal dichotomy.
• How and when did the Republican Party transition into the Democratic Party we see today (and vice versa)? - 8 comments, over 2 years old.
• The commenters in this thread focus on the 20th century changes in the Democratic and Republican parties.
• What changes occurred in the last century that prompted the southern states choose to switch majority support from Democrat to Republican? - 13 comments, over 2 years old.
• Some short summaries and a few books for further reading.

I can't directly answer this question, but with three hours that have passed and no replies I'll throw a little bit more information into the mix.

Even as recently as the 1960s, it seems the modern divide we think of was considerably less defined.

The modern democratic party treats the 1964 Civil Rights Act as a monumental victory of their own doing, but interestingly enough the voting record shows that the Republican party supported the bill more than the Deomcratic party did. The GOP's voting record for the act was consistently 80% in favor or more, while the Deomcratic party was in the 60% and 70% range. Further more, the major efforts to skewer the bill came from the southern Democrats, most notably Democratic Senators Strom Thurmond and Robert C. Boyd, who filibustered the bill for 54 days.

CNN, "What you might not know about the 1964 Civil Rights Act

Wikipedia, Civil Rights Act of 1964, Vote Totals by Party

Senate.gov, Landmark Legislation: The Civil Rights Act of 1964

This surprised me greatly when I learned it, and suggests that perhaps the major swing in platform happened later, maybe in the Regan era.