In the law, if it's not true, and they didn't take ordinary measures to check it out--and they're in the business, and are held to some minimal standards--they can be sued for "reckless disregard of the facts."
If the only source is the Clinton campaign, of course, that makes the presumption of reckless disregard even stronger, and then a good lawyer would look for whether the paper actually knew the story was false.
I thought their response looked incredibly weak. The words they didn't say were: We checked this out, and it's true. Just saying, "We couldn't damage his good name because he doesn't have one" is contradictory. They wouldn't have published it if it didn't represent something about his character that wasn't known. The fact that there's open evidence that they've colluded with the Clinton campaign makes the presumption stronger than they're trying to damage Trump.
If that's so, and they know, or ought to know, that the thing's not true, that's libel. Motive + falsehood + potential or realized damage = $.
To add to the evidence of malice on the part of nyt, if I recall correctly, one of their editors said he was willing to go to jail to defeat Trump. So it makes sense that he would be willing to risk a false story and a libel suit.