Fitness

If you're just starting out, you might want to experiment with your grip, but also what grip implement you're using. It could be that the pulldown bar you're using is just bad for your hands. Wide or narrow grip doesn't matter all that much, so don't believe that you must use the standard wide grip pulldown bar.

You could just get some straps, it will mostly take the grip out of the equation.

Calluses tend to make you less sensitive in the long run though, it could be that you'll feel fine in a week or two, even with no changes.

Going lighter for more reps would also ease the strain on your skin.