There's more than just the one non-failure mode to this kind of thing. My method involves taking the time to consider the information gathered up to the point where I decided to stop listening to the person, as if I hadn't stopped listening to them at all. Information that I would've gotten from them after that point isn't affected by my opinion of them, since I haven't heard it (where it would be, if I were distracted by thinking 'this person's an idiot' as I listened to them), and I give as fair of a trial as I'm able to to the rest.
It may also be noteworthy that I didn't judge him for an argument he was making, and I make something of a point of not doing so unless the logic being used is painfully bad. (Tangential realization: That's why activists who aren't willing to have any 101-level discussions with newbies get a (mild) negative reaction from me; discarding a whole avenues of discourse like that cuts off a valuable, if noisy, source of information.)
Tyler Cowen argues in a TED talk (~15 min) that stories pervade our mental lives. He thinks they are a major source of cognitive biases and, on the margin, we should be more suspicious of them - especially simple stories. Here's an interesting quote about the meta-level: