Previous to the current Western Hemisphere Travel Initiative (WHTI) I strongly believe it is a good idea to take your passport with you when you leave the ship and keep it on your person.
For any official use, a photocopy of your passport is as good as the paper it is printed on. If for any reason you miss the ship and need to fly to another destination or return home you will need the original passport or wait an extended time for a replacement or temporary passport to be created. The photocopy may help because it does have your passport number on it but it won't work to cash a check, or get you on an airplane.
You should ALWAYS take the name, phone number and address of the ship's agent with you when you go onshore. That number is printed in the daily program that you receive in your cabin. Copy it and take it with you.
A cruise is different than a land vacation. The hotel is not going anywhere but a cruise ship is. Keep the passport on your person and not in a backpack or tote bag.
The world is a bit different than it was two years ago.