First of all, are you single? This article is for the needy, not the greedy. Romance on a ship is bound to happen naturally if you meet the right person, but how can you help nature take its course? The first rule is to have the right attitude; cruising is casual and comfortable so acting garrolous, impatient or bored is out of place. When you are comfortable with yourself then the most blissful feeling in the world is having the time to relax and enjoy your own private thoughts. You will naturally attract people who will feel comfortable with you, too.
Be approachable. Don't keep your nose buried in that book no matter how riveting it is. You can read at home, so take frequent breaks just to look around at your surroundings. Look people in the eye and smile at them, after all you are all sharing this experience together. A cruise may be a fashion show at night, but during the day just try to fit in. Neon lipstick and hot pants are out, girls, as are string bikinis. Guys, no Speedo bathing suits.
Make the first move. I know how intimidating that is, but remember that on a ship we are all "strangers," at least to each other, and there is nothing wrong with starting a conversation. You needn't say anything brilliant, it is better to start out with a question rather than a simple statement like, "beautiful day, isn't it?" A question invites a response and is an open invitation to the other person to tell you something about themselves. Be interested, look them in the eye as if they are the most fascinating person you have ever met.
It shouldn't take too long to find out their availability, most people reveal themselves pretty quickly. If they are available then get to know them better. If you distinctly get the impression they are interested in you make a date for the next port. Offer to share a cab to that great beach you heard about, mention you might like to do a little shopping, or perhaps have lunch. Unless you are both very famiar with the port it is best to arrange to meet at the gangway at a specific time, or to pick them up at their cabin.
If you are too shy to do that right away, a great trick is to find out if they are on a tour in the next port. Tell them you "think you may be on a tour, but you aren't sure when or what it is." Get yourself to the shore excursion office immediately and get yourself on their tour. Of course, you don't want to tell them what you did, act as if it was all a big coincidence.
A great place to meet people is in the jacuzzi where you have a captive audience. Be sure to dress right. Other places are the cardroom, library or organized activities like trivia, tea or pool games. One person I know swears that the lifeboat drill on the first day is the ideal place, and he actually met his wife there. "Get 'em early," is his philosophy. Remember this... cruising is a couples experience to a great degree. If you are single and lonely then so is that cute person you are attracted to. Do them a favor and say hi.