A Harris Poll in June 1965 asked a number of questions about civil rights. This survey occurred a few months after the Selma-Montgomery march, and while the Voting Rights act was moving through Congress. That is, the movement was still basically about ending legal segregation and opening up voting rights, especially in the South. President Johnson was popular, with approval ratings of 65-70 percent. In the Harris survey, 19 percent said he was doing an excellent job, 45% said "pretty good," 21 percent "only fair" and only 9 percent "poor." On "handling civil rights and the race problem," people were positive but less enthusiastic: 15% excellent, 38% pretty good, 24% only fair, and 16% poor.
There was a question about the pace of "things that have been going on lately with Negro rights": 40% said "too fast," 27% "about right," 16% "too slow," with 16% not sure. Another question asked who respondents sided with in "recent show-downs in Selma, Alabama, about Negro voting rights." 48% chose "civil rights groups," 20% the "State of Alabama," 18% volunteered that they didn't side with either, and 12% weren't sure. Another asked about charges of "widespread Communist infiltration into the civil rights movement": 23% said there were "a lot" of Communists in "the Negro groups demonstrating for civil rights," 31% said "some, not a lot," 13% "only a few," and 13% "almost none," with 18% having no opinion. When asked to choose among six statements describing Governor George Wallace, 37% chose a favorable one, 37% an unfavorable one, and the rest said that they didn't know.
If I had to summarize these results, I'd say public opinion was no more than weakly positive. And these figures count black respondents, although they made up only 5.5% of the sample (Harris surveyed only registered voters).