Really the only thing that I personally find offensive in Azumanga Daioh is this widely reviled recurring character called Mr. Kimura, who "became a teacher because he likes high school girls". He appears in episodes 1, 4, 6, 7, 10-17, 20, 21, 23, and 26, I think in most of these episodes exactly once — but obviously with a character description like that, even a very brief appearance would leave a bad taste in my mouth. Kimura also briefly appears in the intro — the intro is incidentally also where that "skirt getting blown up" picture is from, that's actually the character getting launched into the air and falling.

Nevertheless, Azumanga Daioh is rated G~PG in most jurisdictions around the world. I've watched the show cap-a-pie with my boomer mom, a bit with my brother, and my mom even wanted to show Azumanga Daioh to an even older relative of ours, too. So Azumanga Daioh is, like the planet Earth, mostly harmless. I have sometimes thought about torrenting Azumanga Daioh just to give Mr. Kimura the ol' treatment, because without him, the worst the show gets is just a few eyeroll moments or maybe specific triggers for some people. You can check IMDb's parents' guide or DoesTheDogDie for more information.

By the way, the "manga cover" I think you're referring to is actually the cover of Dengeki Daioh magazine's August 2001 issue. Probably specifically the picture used at the top of the Wikipedia article for "Dengeki Daioh". So while it is plenty gross that that is in fact official artwork, it is not actually one of the manga covers. Those covers look like so (from eBay):

Edit: removed a gif in case it's an epilepsy trigger