I'm not sure if they had recently rehearsed together at all before they played, or at least this one song, but I've heard many campfire renditions which were far superior to this one.
One issue may have been that a sound check wasn't performed. Ms. Baez' voice soared over everything, the guitars and instruments were patchy, rising and fading as the song progressed. Mr. Kristofferson's voice was barely audible.
Another issue was that both seemed to not be sure who should be leading the vocals, or whether they should just sing in harmony. Their lyrics were often not in sync, with Mr. Kristofferson trailing after Ms. Baez, and Ms. Baez then stepping back and allowing him back in, only to then barge in and drown him out and leave him behind again. Mr. Kristofferson may have been too deferential and allowed her to take the lead, continuously looking over at her during the performance.
He might have been wishing this wasn't the take that was going to be broadcast. Both were appearing in advance of a performance later in NYC. I don't think they sold many more tickets with this showing.
I've heard many memorable performances on Letterman that introduced me to new bands, such as by The Joy Formidable (http://www.rollingstone.com/videos/new-and-hot/the-joy-formidable-whirring-on-letterman-20111007), TV on the Radio (http://www.youtube.com/watch?v=73qBnuzrjx0) and Flight of the Conchords (http://www.youtube.com/watch?v=84EoBQfdrb0), they took this opportunity to blow the audience away and gain new fans. Veterans such as Ms. Baez and Mr. Kristofferson shouldn't take their dwindling audience for granted.