It was Democrats who filibustered anti-lynching legislation for 100 years. The Republicans have nothing to apologize on that score.
Republicans as a party? Yes you are right. However, we must also acknowledge that a large part of those states who are now conservative, were, at the time of the civil rights era, dixicrats. They were not the donks of today. I post this as someone who was raised in AZ by regan democrats. I have voted Republican all of my life, starting with Reagan in 84. My mom and dad have not voted for a donk at any level since '84.
I live in NC now, and can tell you that the south has really reformed it's thinking since the 60's; but in the 60's quite a few folks from my neck of the woods were staunchly opposed to civil rights. We need to be intellectually honest with this issue. Some conservatives, even Goldwater, were opposed to civil rights in the 60's. Now, the GOP did not, and has not become a racist party. Only one Democratic Senator who voted against civil rights shifted allegiances to the Republicans. That Senator, Strom Thurmond, later renounced his segregationist past and voted for the Voting Rights Act of 1980. Most democrats who opposed civil rights such as Albert Gore Sr., J. William Fulbright, and future Democratic Senate Leader (1977-1988) and former KKK member Robert C. Bryd (who democrats now call the conscience of the senate) remained Democrats. The vast majority of southern segregationist did not become Republicans.
Segregationists created the Dixiecrats and stayed with their party of origin, the Democratic Party, when the civil rights movement succeeded. Republicans did not make gains in the south till much later. One can argue, and I do, that segregationists were voted out of office by "the new south" conservatives after the major realingment that nixon started and is just now becoming cemented with the south becoming "red".
As to goldwater's vote, and state's rights in general, both are (and were) valid conservative arguments (AUH20 was not approving "seperate but equal" on a moral basis, but opposed the bill on the grounds that its public accommodations section violated people’s rights to do business with whom they pleased). Brown v Board was the right "idea" enacted by the wrong branch of goverment. I think that eventually, America would have realized on it's own, and voiced through the balot box, that jim crow was bad, on a state by state basis. We would have the same society with no silliness like affirmative action, and resulting judicial tyrany (the courts finding new "rights" for everyone under the 14th). It would have taken longer, but given our current political state of affairs, better.