I have never been to Grenada, Bonaire or Aruba so I can't really speak to those. But I never fealt threatened in St Thomas, or Dominica. So is it safe?" I would say as safe as most places in the states. The fact of the matter is that there is crime no matter where you go, but if you use the same caution you would walking around any unknown city in the States you should be fine. The cruise islands depend on the cruise ships stopping in their ports for the income. The ships would stop coming if the tourists were getting robbed all the time - take the recent occurrence on St Kitts for example. The cruise ships stopped going there for a few weeks after a bus of tourists were robbed. This makes the local official spring into action to 1) catch the thieves and 2) put protections in place to keep it from happening again.
So, if you stick to the well traveled areas and use common sense you should be safe enough.
Carnival Cruise Lines:
Fantasy 90, Celebration 95, Victory 05, Conquest 05, Spirit 6/06, Freedom 6/08, Holiday 7/08, Victory 6/09, Valor 9/09, Dream 6/10, Splendor 8/10,Conquest 8/11,Magic 11/11,Triumph 4/12,Spirit 7/12,Freedom 11/12, Triumph 2/13 - The fire cruise
NCL Dream October 2006, RCCL Mariner OTS June 2007, Celebrity Silhouette 2013