Divide and Conquer
First, we must break away from the paradigm of the Coast Guard as a defensive, domestic, security apparatus, which patrols America's coasts, lakes, and rivers, and consider it as a brown-water navy, which takes as its purvey all of the lakes, rivers, and coasts of the world. The Navy can then be focused as a blue-water navy whose job is to fight enemy ships--in essence, to keep the seas clear so the brown-water navy can do its work.
Second, in the last few wars, the Marines have been used primarily as light infantry, because their speciality, amphibious landings, are not often needed by the United States (between helicopters and hovercraft, amphibious landings have become rather routine). However, the Coast Guard could definitely use some Marines. Boarding ships, the pre-ironclad purpose of the Marines, is only practiced by the Coast Guard (plus, they could continue their anti-piracy history "to the shores of Tripoli"). Riverine war, furthermore, requires the ability to project force onto the banks of the river. Such small scale amphibious landings (a squad, or at most, a platoon at a time) would fit very well with the Marines' structure and professionalism, since it would rely very heavily on junior officers and non-coms.
River crossing, the most hazardous manuever in a ground war, would also be a Marine responsibility. The Marines and the Army have different tactics. The Army tends to husband its troops carefully, since a soldier you don't lose now is one you have for a later battle. The Marines, by contrast, are willing to accept high casualties to achieve their objective, because if they don't get off the beach, they're all going to die. Crossing a river fits the tactical nature of the Marines far better than it does the Army.
There are three environments that we deal with: land, which the Army handles, water, which the Navy handles, and the land/water mix of coasts, lakes, and rivers, which both the Coast Guard and the Marines handle. The Coast Guard is mostly water, the Marines are mostly land, but together, they would be able to dominate that environment more effectively than either could alone.