Is Cartman an anti-semite because he hates Kyle and knows Kyle is a jew, or does he hate Kyle because he is an anti-semite and knows Kyle is Jewish. Did Cartman become an anti-semite because it fuels his hatred of Kyle, or does he hate Kyle because he is an anti-semite?
Cartman hated gingers too, which Kyle is one. So I’m inclined to believe the former.
I was always under the impression that Cartman hated everyone, with Kyle just being in an especially easy category to give shit.
Kyle is not a ginger: he has red hair, but no freckles. This makes him a “daywalker.” 
I think that Kyle’s heritage is just something Cartman can focus on. Cartman hates everyone (most of the time), but Kyle calls him out on his stupidity more than anyone else and so Cartman often focuses his anger on Kyle. Kyle being Jewish allows him to go after Kyle for his “different-ness.”
If Kyle wasn’t Jewish, or if there were no Jewish kids in town, it’s hard to say whether Cartman would care about “jews” at all.
To fully understand the hatred you have to know that Cartman and Kyle were actually switched at birth. Kyle’s red hair is due to carrying Tennorman genes, and Cartman is really a self loathing Jew in denial who gets his fat from being Mrs. Broflofski’s biological offspring.
No, this has never actually been revealed on the show or by the creators or anyone official, it’s just so obvious if you actually pay enough (too much?) attention. So no spoiler boxes.
Join us next time folks, when we discuss the evidence that Butters is actually a clone.
Please, please get a job writing for Matt & Trey!
I think the anti-Semitism came first, and is probably an attitude he hears from his mother. My evidence: in the first season Halloween episode, his Halloween costume - which I assume his mother made him, back then the main characters were relatively ordinary third graders and not omni-capable super-beings in third-grader bodies - was Hitler. And back then, he didn’t seem to have anything in particular against Kyle.
Having been a fan since the beginning, yeah that’s true, he did say his mom made it for him. But in another early, first season episode he also calls Kyle’s mom, “a nosy Jew!” so I think the intent was that it was Cartman himself all along. Indeed, Trey & Matt said they based him on Archie Bunker, a funny, buffoonish, folksy racist.