A majority of the noise from an outboard motor is generated through the engine housing. There are a variety of solutions to the problem, the simplest of which is to wear ear protection. Before considering any of the steps outlined below, ensure that the unwanted noise isn't being generated by the boat's passage and the resulting wind.
Lower the engine so that the prop isn't exposed when the boat is operated at higher speeds.The transom at the back of the boat may need to be cut down a little to do this. Remove the engine first, using a mobile engine hoist and lifting rings for large outboard motors. Alternatively, try changing the props on the engine (as some, such as ventilated props, run more quietly than others).
Fit the engine with a shroud. A number of manufacturers produce cowling designed to reduce engine noise. Some of these use foam as a sound dampener. If this is the case, consider removing the foam and replacing it with a better sound dampener, such as that available from car audio stores.
Build a noise-shielding enclosure. This can be as simple as erecting a piece of plywood above the transom, so that engine noise is reflected away from the driver. Measure the width of the transom and the height to which the engine extends above it. Cut a square of plywood to these dimensions and mark out a space to allow the handle of the engine to come through. Screw the plywood to the transom.
Consider changing engines, if the above should fail. Four-stroke engines are less noisy than two-stroke outboards. Engines from different manufactures are associated with different amounts of noise. Suzuki and Evinrude both produce relatively quiet outboards. Be aware, however, that in some cases a quieter engine is also less powerful.