If you are meeting the ship to sail to the Caribbean, Brooklyn is perfect. It is quieter than the Manhattan piers, and close enough to get to via taxi. Also, if you arrive in the New York area a day or two prior to sailing you get to see the Big Apple and two vacations in one.
Dominica is not known for its beaches. Instead we walked around town to see how the locals live. We were warned about crime, but did not feel at all threatened or uncomfortable. Everyone was friendly and helpful. We even bought a passion fruit drink from a street vendor that was delicious, and wound up sitting and talking to the folks nearby who were all from the island.