Act friendly and be positive. People are drawn to outgoing and happy individuals. Greeting each person you run into in a friendly manner will start conversations that could lead to more meaningful friendships as the cruise rolls along.
Sign up for onboard activities like shuffleboard, talent shows, wine tastings and poker tournaments. Check out your cruise ship's activity schedule for things that interest you and attend those activities to meet like-minded people. The activity gives you a natural way to start a conversation.
Attend dinner or lunch in the main dining room of your cruise ship. Seating in these environments is usually mixed, meaning you will end up sitting at a table full of people you have never met. This can seem intimidating at first, but sitting with different people at each meal helps you make new contacts and talk to people outside of your normal comfort range.
Hang out in areas of the ship that would contain the sort of friends you would like to make. The ship's bar, for instance, will often have young couples or single people, while the dance floor may skew older. Looking around the ship and determining what sort of person frequents which area can help you put yourself in the right place to meet people like yourself.