The shore excursion description will tell you whether or not you need your passport. Chances are you won't need it. We did excursions in northern Europe and just had to take our boarding pass with us.
We always keep our passports on the ship, in the in-room safe if one is provided, or hidden somewhere. We've been on a couple of ships that actually collected our passports when we boarded, and gave them back to us the night before debarkation.