Interesting challenge here, journalistically. Roger Cohen's column in today's Times made the point that "fact-based journalism" is what we used to call "journalism" and that there shouldn't be any other kind. Of course he was referring to more pressing issues than washed coffee, but this article is a good example of the erosion of evidence in favor or unfounded belief. The journalist goes out of her way to say that empirical evidence doesn't support this crackpot's claims, but she's still giving him a platform. And as we've seen, it's the platform that matters and not the message. Not that I'm comparing bogus claims about coffee to the START treaty necessarily, but I think this article points to the bigger issue that controversy is so much sexier than facts. If only the facts mattered here, there wouldn't be an article to speak of.
I think the only reasonable tack for journalists to take is to start writing headlines that say, "Trump lies about X". It lacks the decorum that most respectable publications would prefer, but these are not normal times and normal solutions aren't going to get us anywhere. If CNN, NYT, and NPR all started just routinely beginning stories with, "Today the administration lied about such-and-such," he would implode in a heartbeat. It's not a qualitative judgement if it's true, so call a liar a liar already. The naked truth might just liberate a bunch of other journalists from their need to be "balanced."