It all depends on which cruise line you are travelling on. NCL has you change your clocks to coincide with local time. It makes it less confusing when going ashore. When we were on Carnival a few years back, they had you stay with Eastern time. Unfortunately, poeple going ashore expecting to shop for an hour before the shops closed were dissapointed because the stores were closed.
You should stay on ship's time, but also be aware of the time on any island not in the same time zone. Also beware of some countries that don't honor daylight saving time.