"Geography Was Far More Predictive of Voting Coalitions on the Civil Rights [Act] Than Party Affiliation"

"In this case, it becomes clear that Democrats in the north and the south were more likely to vote for the bill than Republicans in the north and south respectively. This difference in both houses is statistically significant with over 95% confidence. It just so happened southerners made up a larger percentage of the Democratic than Republican caucus, which created the initial impression than Republicans were more in favor of the act.
"Nearly 100% of Union state Democrats supported the 1964 Civil Rights Act compared to 85% of Republicans. None of the southern Republicans voted for the bill, while a small percentage of southern Democrats did."

In a 2013 Guardian article, Harry J. Enten answers the question, "Were Republicans really the party of Civil Rights in the 1960s?" 

"Overwhelming Evidence Showing the Central Role Racial Politics Played in the Republican Party's Rise to Power After the Civil Rights Movement"

"Buoyed by the surprising strength of Workman's campaign, South Carolina’s junior senator, Strom Thurmond, abandoned the Democratic Party in 1964 and joined the Republicans. Sixteen years earlier, Thurmond had left the Democrats briefly to run for president as a 'Dixiecrat' on the States’ Rights ticket. His surprise announcement in 1964 signaled the start of what political scientists call the 'great white switch.'"

Sid Bedingfield at The Conversation discusses the rise of the Republican Party in 1960s South Carolina.