A stolen US Passport is a valuable item. Guard your passport as closely as you guard your wallet. The most likely place for you to misplace your passport is while you are going through security in the airport. It is the single most important item you have with you. During airport security agents will ask to see it at least 4 or 5 times before your board the plane. Never let it leave your sight. You cannot cross any international border without it and that includes coming back to the US.
There are times when you will need your passport: at the airport, going through customs, checking into the hotel, boarding the ship. You won't need it while you are walking around Rome or in any of the port cities. Your hotel room will probably be equipped with a room safe. Your ship cabin will definitely have a safe. Use it.
When we arrive at each port the ship authorities will tell you exactly what documents are needed to leave and to reboard the ship. In most cases it will be your Costa Card and a photo ID ... not your passport. If the passport is NOT needed then you should leave it locked inside the safe.
Israel has high security standards. A day or two before we reach Israel the cruise line will collect all of our passports and present them to the Israeli immigration officers while we are at sea. Once the passports have been screened and processed they will be returned to us before we dock in Ashdod. You will notice high security throughout Israel. Metal detectors and other security measures are used everywhere there are crowds of people. There is normally a security check before you enter a shopping mall, public park, etc.
Greece is a member of the EU and we are arriving on an EU vessel so no additional check is necessary.
Visits to Turkey normally require a Visa (different from a passport.) There is an exception for cruiseship passengers whose stay in Turkey does not extend overnight. The cruiseship will take care of the paperwork so just listen to the ship announcements for details.
You should make a copy of the photo page of your passport and keep it in your wallet or your carry-on luggage. If you lose your passport the copy won't do you any good at the airport or in the customs line. It will, however, help when you report to the US Embassy or Consulate and attempt to get a new one.