Honestly, I like anime the same way I like all art forms. They need to be done correctly, or at least with a sense of artistic expertise and originality, or else I despise them. This isn't just for anime, I hate when realistic art styles are done badly. From making explosions and contrast bland (look at old WW2 films showing the bombing runs at night, look at the contrast. I want to see someone do that correctly. it's a beautiful (albeit horrific) culmination of bright orange (or white, since most are black and white) and jet black.) to not having realistic gameplay (bullets don't travel in straight lines). When you have a certain artstyle, you tend to constrict yourself to a certain gameplay style as well. You can try to do a different style, but you risk making your game seem odd and very out of place.
I do enjoy some anime, particularly with things made by Miyazaki, with Spirited Away being my personal favorite, as it has fantastical elements that were played so straight, yet felt so...right. I've only seen this with one other medium, and that was with Morrowind, which had a much different artstyle, yet had a similar concept of having fantastical elements played straight. That was also why I hated Oblivion's artstyle, as I felt the "cartoony" look was toxic to the environments of Cyrodiil.
However, anime seems to follow certain...issues. I'll split this into two parts. When I see anime fall flat on its face, it tends to follow these two, that being "animu" and being very generic. "Animu" is a stereotypical art style with anime (and is often considered to be anime, though I would consider it BAD anime) They tend to have tell-tale signs.
-mouths being "off to the side" of faces
-very sharp chins
-"spur of the moment" emotional scenes, where characters have rather disproportional structures, often having a "O.O" face
-faces of characters are often very geometric. This can be done as a specific art style (often making it seem like an older comicbook), but in this case, it is rather bad looking, and looks rather cheap.
-Faces, and even profiles of characters, look rather the same, with any nuances being hairstyle, palette choices, and weapon variety
Now, notice that these aren't the requirements to be an "animu", but most "animu" tend to have these. Another issue is how generic bad anime tend to be, often having a very...well...wangsty storyline, with mary-sue characters and drama that fail to reach the curriculum that most western drama follows. Now, often for me, ignoring the bad anime, often the main issue with anime is they tend to be very...eastern. From rather fantastical elements that just seem to be accepted, to monsters and creatures that follow ancient Shinto religious elements, which to the western viewer, is often so alien and Other, that it looks like an Eldritch Abomination.
I hope that helps...