In an editorial about the GOP primary that was written in the wake of the interview, the Courier-Journal noted with concern that Paul "would favor dismantling several federal departments including Commerce, Education and perhaps Agriculture."
As for the Civil Rights Act, here's the relevant portion of the interview, which was posted on the paper's website:
Questioner: Would you have voted for the Civil Rights Act of 1964?
Rand Paul: I like the Civil Rights Act in the sense that it ended discrimination in all public domains and I'm all in favor of that.
Rand Paul: You had to ask me the "but." um.. I don't like the idea of telling private business owners - I abhor racism - I think it's a bad business decision to ever exclude anybody from your restaurant. But at the same time I do believe in private ownership. But I think there should be absolutely no discrimination on anything that gets any public funding and that's most of what the Civil Rights Act was about to my mind.
Questioner: And then it was extended by most to most localities to include all... Would you be in favor of just local-
Rand Paul: On a local basis it might be a little different. The thing is I would speak out in favor of it. (pause) I mean, I look at the speeches of Martin Luther King, and I tell you I become emotional watching the speeches of Martin Luther King. I loved him because he was a transformational figure, but he was a believer. what i don;t like most about politics is almost none of them are believers and he was a true believer and he fought government injustice, and those were governmental rules and laws that forbid people you know from riding the bus or sitting in certain parts of the bus or drinking water form public fountains. All of that should have ended and I think it was a great occurrence that it did.
Questioner: But under your philosophy it would be okay for Dr. King to not be served at the counter at Woolworths?
Rand Paul: I would not go to that Woolworth's, and I would stand up in my community and say it's abhorrent. um... But the hard part, and this is the hard part about believing in freedom is, if you believe in the First Amendment, for example, you to, for example- most good defenders will believe in abhorrent groups standing up and saying awful things, and we're here at the bastion of newspaperdom (sic) and I'm sure you believe in the First Amendment, so I'm sure you understand people can say bad things. It's the same way with other behaviors. In a free society we will tolerate boorish people who have abhorrent behavior, but if we're civilized people we publicly criticize that and don't belong to those groups or associate with those people.
Questioner: But it's different with race, certainly a hundred years, discrimination based on race was codified under federal law.
Rand Paul: Exactly, it was institutionalized and that's why we had to end all of the institutional racism and I was in favor completely of that.
Here's the video. The exchange above comes around the 1-hour mark:
So Paul makes it crystal clear that he opposes banning discrimination by private businesses.