In short, it depended on the white Southerner. The tendency here is to conflate white Southerners with white Southern activists. The majority of Southerners during the Civil Rights Movement were not actively involved in resisting. Of course, this is due to the success of segregation, where some white Southerners rarely had a chance to interact with African-Americans. Jason Sokol's There Goes My Everything is a particularly relevant book for this question. Essentially, he argues that the CRM changed the lives of every white Southerner.

Some felt betrayed by African-Americans, who were violating a paternalistic relationship that many white Southerners treasured. Others were more open, even inviting African-Americans into their own businesses. Another historian, Dave Chappell, suggests that the Civil Rights Movement succeeded because the majority of white Southerners did not have a moral or religious motivation to preserve segregation. It should be noted that his book, A Stone of Hope has recently been challenged by Carolyn Dupont's Mississippi Praying, which argues that white religious institutions did seek to enforce segregation.

This is just a very brief overview, but it suggests that this question has many historiographical answers. I think reading any of the books I mentioned may deepen your understanding of the time.