It was actually the Civil Rights Act of 1964 that Goldwater voted against; he wasn't in the Senate anymore by the time the Voting Rights Act of 1965 came up for a vote. Goldwater might have opposed that bill too, though he had backed voting rights provisions in the Civil Rights Acts of 1957 and 1960. Goldwater later said he supported all but two sections of the 1964 law as well, but voted against the legislation because he felt those sections were unconstitutional.
That said, Hillary's invocation of MLK does sound a bit like the standard Clinton yarn-spinning, much like when Bill recalled Arkansas church burnings in his youth that apparently didn't happen. While Goldwater was committed to color-blindness like King, in the political climate of the 1960s it would have been unlikely for someone to have been a fervent admirer of both. Goldwater was increasingly critical of the civil rights movement's politics; King likened to Goldwater to Hitler.