All you have to do is search the statement, which is false.
Percentage-wise, in regards to the parties themselves, more Republicans than Democrats voted for, but in terms of the tally of votes, which is what matters, more Democrats than Republicans voted in favor of the 1964 Civil Rights act. In fact if it wasn't for LBJ and them, said Act would never have passed.
The division wasn't along party lines, it was a long regional lines. As usual, the North was in favor, while the South was against.