It had a huge effect. For the past 35-45 years the Republican Party could be reasonably viewed as a white people's party because on a national level they are almost exclusively supported by white people and tend to support policies that would roll back civil rights laws that they see as excessive (affirmative action for instance).

The democrats on the other hand have a much more diverse base of support and almost always seek to preserve and expand civil rights legislation.

This was not always the case.

From around 1850-1950, the Republicans were the party of racial equality (or something approaching that). They were abolitionists before the civil war, supported rights for former slaves after the conflict and could generally count on the support of African Americans.

At this time Democrats for their part could generally count on poor, urban whites in the north east and rural white southerners as their main political bases. Southern democratic politicians usually branded themselves as white populists, had strong confederate sympathies well into the 20th century and often had very open ties to groups like the KKK.

As black people from the south moved into urban democratic strongholds and began to demand civil rights across the country, the National apparatus of the Democratic party decided it would be politically worthwhile to champion black rights, both to keep control of cities and to get the strong support of black voters across the country.

LBJ in particular personified this change when he signed civil rights act of 1964. He knew that support for this would cost the Democratic party the "solid south" but he thought it was worth it to gain minority and progressive support, and also arguably thought it was the right thing to do.

LBJ is quoted saying "I know the risks are great and we might lose the South" which of course they did, but support for civil rights won the democratic party the lasting support of black people and other minorities. The republicans of course took advantage of this, became the new champion of southern/conservative whites.

TL/DR

More broadly Democratic support for civil rights solidified a trend where the Democrats moved away from simple populism and machine politics to a Party defined by support for minority rights generally and trust in government intervention to cure social problems.

Republicans for their part moved from being a patrician, centrist party that was based on the support of well educated, coastal professionals to a party that is based on preservation of traditional social roles and distrust of government intervention which is firmly based among rural whites.

Both these changes had been brewing since at least the New Deal era, but the Civil Rights movement was a watershed.