Conservatives like Star Trek because it promotes the values of meritocracy, hierarchy, authority, and military discipline. None of these are commonly held by people in the American left (especially not the academic left, which tend to be pretty anti-hierarchical, anti-authoritarian).

The people usually referred to as Republicans these days are right-wing populists. They have very little in common with the types of conservatives who loved the franchise in the 20th century. Those older conservatives are almost extinct in the Republican Party today.