Superman did and does fight for social change. In the Superman Archive, he clearly starts out as a populist hero, a champion of the working class, taking on war profiteers, state-run orphanages, crooked boxing promoters and poor mining conditions. Watch Superman lead a party of upper class twits into a mine and then bury them alive to teach them a lesson and tell me that's not a guy who fights the power.
And today he fights against class elitists like CEO (and ex-PRESIDENT) Lex Luthor and monarchic dictators like Darkseid. Compare that to Batman's typically lower class, obviously criminal, more anarchic villains. Superman fights against those who would impose their own version of order on the world, while Batman fights those who would destroy the order HE imposes on Gotham.
And while Matthew's right, it would be morally repugnant for Superman to enforce social change, Clark Kent can and does champion those changes from his job as a reporter for the Daily Planet.
Clark, after all, had a lower middle class rural upbringing and a strictly middle class life style once he became a reporter. Sure, it's a "glamor" career that makes him somewhat famous, I'm guessing he doesn't actually make that much money (Lois might). He might not have Peter Parker's money problems, but Clark almost certainly knows what it's like to worry about the bills.
But Superman is the exception here, not the rule. By nature, a superhero is someone whose unique abilities places them apart and above, sometimes literally above, most of society. That these unique beings then go on to be vigilantes, placing their own personal definition of justice above that of the police and democratically elected government, is elitist, aristocratic, and borderline fascist (I'm looking at you, Batman).
My own feelings on the issue were sketched out here, but I would tend to disagree: Superman is the basic template for all things superhero, including manifestations of class, and while not exactly aristocratic, he is certainly "king" of the castle in the superhero world and in the real world of corporate properties. Also, I'm not sure what a "class elitist" is but I'm not going to pick favourites in a fight between a middle-class vigilante alien who can make diamonds and a capitalist kingpin.
(All this started from a post about class in the future at Legion Abstract, apparently.)
As well, interested readers should check out this article by the great Jeet Heer on Red Son, a mediocre Superman comic from a few years back that re-imagines The Man of Steel as a communist.