Graphics can play a part on whether I buy a game at release at full price or not, but it is not the only factor. Most of the time graphics isn't even a major factor. There are exceptions, I am usually very harsh on the art styles of Visual Novels. If I don't like artstyle, unless its highly regarded I won't bother with it. I will usually give a flawed game a try after a price drop/ Ebay snipe as long there is some part of the game that interests me, gameplay, story, premise, characters, etc. I have been an anime fan since I was around 12, so yeah, I like anime styled graphics. I also like realistic graphics as well, games like Heavy Rain and Uncharted 2 look amazing. The Witcher 2's graphics look phenomenal and I am going enjoy seeing how well my upgraded computer measures up to it performance wise.
I don't care what a game looks like so long as it is fun.
One possible exception would be some of the early 3D stuff (like PSX era) in which characters were nothing but a handful of jagged polygons. That really hasn't aged well. I can play older sprite-based games just fine, but there is something about the early years of 3D games I can't stand.
I second that. I can play SNES or Mega Drive games with no problem, but anything on the PlayStation that is 3D with sprites... urgh.