How was the conflict between supporters of a strong federal government and champions of states rights characterized then, as opposed to now?

Until the 1960's, "State's rights" was mostly a rallying cry of the South for preserving racial segregation. The unanimous Warren Court school integration decision (in 1955) and the civil rights and voting rights laws passed under the LBJ presidency in about 1965 changed all that.

State's rights issues today are mainly things like oil drilling leases and immigration control.