OPs are usually the things that I use to get a first impression about the series.
Shugo Chara, Bokurano, Prism Ark, Kyouran Kazoku Nikki, TTGL, Heroic Age, Bartender, Hidamari Sketch, Magical Pokahn (I lol'd a lot with the rest of /a/ at the difference in tone of the OP and the actual show), Code Geass, Shigofumi, and a number of other shows had decent OPs that convinced me that the shows were initially worth watching.
Of course, that doesn't say much about the actual shows, like Bokurano and Prism Ark (Oh god, AEN's subs were so horrible and shitty that it was painful sitting through them). The OPs just got me hooked, but there wasn't enough content to hook me on. Sometimes, the EDs compensate for the terrible OPs if they're good enough (ie: Vampire Knight, Kamen no Maid Guy (WORKGUYWORKGUYWORKGUY!!!!)).
I can't think of a show with a terrible OP that I decided to watch through. Shows like Night Wizard, which have terrible OP/EDs IMO, are still in the queue in the back of the line behind the more recently released anime and stuff that I'm planning to watch soon because I just never feel like watching them.
tl;dr - The OP/EDs probably are a huge influence on what I decide to watch.