Travel During the Low Season
Traveling during the low season makes everything cheaper. There are less tourists so hostels are not crowded and that's when you can find the best deals. The low season in the Caribbean is from May to October but keep in mind that June to October is also the hurricane season. If you are going south of Dominica, you are very unlikely to be hit by a hurricane. However, if you are going north from there, keep an eye on the weather reports. Showers are normally late in the afternoon with clear skies the rest of the day.
Bringing a tent or a camping hammock is one of the best decisions you can make to save money. Some islands are better than others for camping. I found campground areas in Martinique, St Lucia and Dominica with all the facilities needed and very affordable camping fees. Some islands may not have campground areas, but they have plenty of beautiful and isolated beaches. Safety might come to your mind when planning to free camp on the beach, but it's very unlikely that you will have any problems. Just try to find a beach without any town nearby. I really enjoyed free camping on the beaches of Carriacou, The Grenadines and Barbuda. And if you are in the right season, you might even spot leatherback turtles at night!
Eat What The Locals Eat
Whatever it's produced on the islands is cheap. The options are not very diverse and rice and beans are always on the menu. Fruit is a great option. Mangoes, bananas, papayas and other tropical fruit are plentiful and delicious. And don't forget about drinks. Beer and rum are surprisingly cheap. Whenever you are in a big town, try to visit the local market. There are normally plenty of stands around them where you can get cheap meals and have a chat with the locals. It's my favourite option.
Use Boats When Possible
Cargo boats are one of the cheapest options but they are infrequent and don't go to every island. Also, it's very difficult to find departure dates or any kind of schedule if you are not on the island already and go to the port in person. Local boats tend to be cheap but they are only available on short crossings. It's a great option to travel through Grenada and The Grenadines group as the islands are close to each other. The quality of the boats varies and they absolutely never run on time, so you will need to be very patient. Finally, you always can try to hitch a ride on a sailboat!