Back in the day, Carson, Leno, et. al. would make fun of both sides. Now, most of it seems to deliberately target one side, and in a heavy-handed way. Jimmy Fallon is probably the late nighter most akin to the past generations of late night shows, in that he makes fun of both sides, and the political jokes aren’t the main thrust of his shtick.
Of course, when the president is so thin-skinned about taking a joke that he even threatens to go after Fallon, he makes himself an easy target.