B737 Captain reported the yoke nose trim switch on both sides would not command nose trim down during climb out. Flight crew continued to destination airport and landed safely.

Narrative

During climb out the yoke electric nose trim switch no longer would command nose down trim when actuated. Nose up would work and the autopilot actuated stabilizer trim would also function. After confirming with the FO that he was getting the same results; I turned on the auto pilot transferred the controls to the FO and began to run the stabilizer fail QRH. Before completing the checklist; I contacted dispatch to get a Maintenance Control patch to see if they have any suggestions as completing the checklist would remove the auto pilot availability. They responded with no additional advice outside of the QRH; so we completed the checklist and I had the FO hand fly; while I briefed the flight attendants and the passengers. Once completed we had already begun our descent into ZZZ; I [requested priority handling] transferred the controls back to myself and continued with the descent and landing. Once clear of the runway I had fire rescue do an inspection to verify everything was ok externally then taxied to our gate. During the debrief with FO it became clear to me that I didn't do a good enough job of adequately sharing my thought process on the timing of the checklist completion and when I would [request priority handling] with ATC. He was ok with the pace of my actions and the results however; I would have made him more comfortable during the event if I had explained the why in the moment; before performing the tasks.