Of course he'd support it today.
What is your reasoning for LBJ being on board for gay rights? An issue like this would make his egotistical head explode with rage. Irony is you had Midwestern conservatives (like me), Northeastern libs and others for civil rights against the southern one-party democrat/Dixiecrat racists.
It's 2014. McCaskill, Donnely., Hagan, and Heitkamp support gay marriage, and so would he.
As for civil rights, northern Democrats were more likely to vote for it than northern Republicans, and southern Democrats were more likely than southern Republicans.
http://www.theguardian.com/commentisfree/2013/aug/28/republicans-party-of-civil-rights