I can't answer the passport/ID question but given the relationship between the Bahamas and the UK I'd think an official photo ID would suffice for you.
As to the other question...indeed, Nassau is becoming more affected by crime. I have not heard of cruise lines dropping this port, and the Bahamian police have stepped up their efforts in the tourist areas. If you stay on the beaten path you'll be fine. It is the tourists who wander off into the back streets who find trouble. I know there was a famous incident a few years ago where cruisers visiting the Queen's Steps were robbed, but this type of thing is not common.
Personally I have had enough of Nassau and I stay on the ship when there.