Some say that civil rights would just get passed in 1965 after a Democrat landslide, but I doubt that. First, about 80% of Republicans supported civil rights, even more than the Democrats so getting rid of Republicans won't be much help since it was a bipartisan issue. And secondly if civil rights failed once the President wouldn't want to waste political capital on it again, and probably focus instead on the War on Poverty.
First of all, if the filibuster succeeds, it will probably succeed very narrowly. So if the election produces even a couple of changes in the pro-civil-rights direction, that would be enough. And in OTL it did: In New Mexico, Edwin L. Mechem (R) who voted against the law was replaced by Joseph Montaya (D) who supported it. In Tennessee, Herbert S. Walters (D) who voted against the law retired and was replaced by Ross Bass (D) who "was the only Representative from the rural
South to vote for the
Civil Rights Act of 1964,"
https://en.wikipedia.org/wiki/Ross_Bass (It is also possible that having been safely re-elected in 1964, Albert Gore, Sr. will now vote for the bill; after all in OTL he voted for the Voting Rights Act of 1965 and the Civil Rights Act of 1968.)
Second, while it is true that *in OTL* most Republicans in the Senate voted for the bill, in a scenario where the filibuster succeeds, there will presumably be at least a couple of more Republican opponents. (It's not very likely that the extra votes to defeat cloture will come from Democrats--all the southern Democrats in the Senate except Yarborough opposed the bill in OTL anyway.) So if those Republicans or some of them are defeated in November, it could make a difference.
Third, it will simply not be politically possible for JFK, having committed himself to the bill and almost succeeded in passing it in 1964, and with the bill being an issue in the 1964 election as one of the major differences between himself and Senator Goldwater, to pass up the chance to enact it now that Goldwater had been defeated and the votes were there. The political damage in having pressed for the bill in the first place (basically, alienating the South and especially the Deep South) will already be baked in, anyway.
Fourth, why is this issue always presented as the bill in its 1964 form versus no bill at all? If it's only one or two votes short of passing, it could be weakened slightly to attract the extra one or two votes. (For example, there could be exemptions for "truly local" businesses with only a very indirect effect on interstate commerce, etc.)
Fifth, even in the very unlikely event that Congress does nothing, the Supreme Court within a few years will interpret the Civil Rights Act of 1866 to ban private discrimination:
https://www.law.cornell.edu/supremecourt/text/392/409 This was a 7-2 decision, so there seems little reason to think it would have turned out differently it JFK had lived.