I have two stances on this.
First is for when I'm playing older games. In this instance, the graphics don't bother me. If I'm playing FFVI, then I know I'm going to have blocky sprites, but that's actually part of its charm. It was made God knows how many years ago, I'm not going to hold graphics against it - that was then and this is now. So I won't be stopped from enjoying something because it's old and not incredibly shiny any more, because times change.
New things. Shoddy graphics still won't stop me playing something, but I'm more willing to call a game out on it. In this day and age, there's very little excuse for having poor quality aesthetics. Look at God of War, Mass Effect, Gran Turismo. It can be done, so do it. Case in point here is Dragon Age. I love that game, I really do. But when I play it, there's a niggling bit of my brain that says "look at the hands on those models, surely they could have done better than that?".