It makes me leery, sure. I'm not a person who enjoys multiplayer typically - although some team based multiplayer vs. AI is tolerable and occasionally even enjoyable to me, I completely detest multiplayer that has the option of players talking (for obvious reasons) and I'm ambivalent trending towards dislike of multiplayer that pits player against player even silently.
There are shining exceptions. Journey is the only one that springs to mind. Of course that is the point of the whole thing and there are no annoying voices of screeching profanity to avoid, so that's a big help.
Overall though I do not engage the multiplayer. I do not desire games that are single player to offer a multiplayer mode. I particularly dislike multiplayer that impacts or is made a component of the single player. And yes, this extends to Mass Effect 3's multiplayer - which, while I did not hate it with the intensity of the burning of a thousand suns, I do find extremely tiresome and aggravating in the sense that, even on my millionth play through I will have to participate in if I want all my work in the single player amassing resources to be fully counted. I usually have my fiancé do my readiness before the "point of no return" at the end. It was a little bit cool the first time, mildly engaging the second time, but by the third time I ardently wished I could just "lock in" my readiness already and be bloody done with it. The novelty has long since worn off.
I certainly didn't bother with multiplayer in The Last of Us - hell, I didn't even know it was there until I saw a discussion about it on these boards.
Call of Duty I enjoyed the "on the couch" multiplayer, with my fiancé, because it was nice to play together. Local multiplayer is something I look for in games (Lego games, the older CoD games, various dungeon crawlers, etc.) but Online multiplayer... no. I don't think I need to even bring up CoD multiplayer online's total lack of appeal for me, considering my previous comments on the hatred of voices screaming profanity constantly.
I even turn off the online in Borderlands 2 because I hate it SO much. I hate even MORE what using it did to my fiancé's play style, even when he's just playing with me. I flat out refused to play it with him after he spent time with the online population, because he kept treating me like I wasn't even there and just running around starting and doing quests without the usual fun conversation and inclusive working together we enjoyed before, when it was just us playing together pre-online people experience.
I wouldn't say - though - that it turns me off of a game to know it has multiplayer added on. I mean, a lot of games do and some of them are decent or even good and I don't want to miss out on them out of prejudice against one particular element. I do wish it wasn't so damned popular and commonplace now though. I wish it would find its niche and stay there instead of being something tacked on to absolutely everything these days. Very much so.