I don't think crocodile tears are the right term. Many adults cry when they are frustrated and angry. Not that we want sympathy, we are just feeling emotion that comes out our eyes. The best thing to do is not do anything. Continue the argument/discussion/debate as if no one is crying.
If someone cries because they don't like your answer to something? I would still do nothing about the tears, and reevaluate the friendship. I have had experience with that, I pretend they aren't crying, or, if the conversation is interrupted to the point of stopping, I ask coolly "are you finished" and continue with my last point.
People cry for all kinds of reasons. I think the trick is figuring out why. Is it because that is how emotion is released for them? Did something really sad happen and as another PP said, the hole in the sock is devastating today, or is someone trying for emotional blackmail? With out a specific example, it's hard to really give a concrete answer.
Hopefully the crier is someone you know well, and can gauge why crying is happening. If it's an acquaintance, ugh, I find that incredibly awkward and I have just stood there like a fool, not knowing where to look, what to say, or if I should just leave.