Acceptable Targets: A little person is mocked and threatened in At The Circus. It works because it makes Groucho seem cowardly and weak instead of cruel.
Upper class people don't generally come up with the longer end of the stick either. But then again, few people get treated well by the characters the Marxes play.
Of course, a lot of that comes from the Depression Era roots of the act; The Three Stooges and other comedy troupes of the time did the same.
Once-Acceptable Targets: "Then the Headstrongs married the Armstrongs, and that's how Darkies were born" -- Groucho in Duck Soup. Also, Chico's portrayal of an Italian immigrant, Harpo's original "Irish bruiser" character from his vaudeville days, etc. Hey, it was the '30s!
The "that's how Darkies were born" thing was actually a pop-culture reference: it was a play on an incredibly racist song that was popular at the time, "That's Why Darkies Were Born": the lyrics of which basically said "someone had to be slaves and have crappy lives -- so it's you" It could be read that Groucho-- himself a Jew, and probably quite familiar with racism-- was ridiculing the warped logic of the song.
Of course then there's the scene in The Big Store where they have black actors literally picking their cotton in-store...