The US expanded its presence and influence in the Caribbean in a number of ways.
The most obvious way was through war. In the Spanish-American War, the US took possession of Puerto Rico and also took unofficial control of Cuba. (Cuba remained independent, but the Platt Amendment gave the US a great deal of control over it.)
A second way was through what is known as "dollar diplomacy." This was a policy of using American money (mostly via investments0 to try to get more control over Caribbean nations. The policy was also meant to stabilize the economies of those countries with superior American know-how.
A third way of expanding influence was through the Roosevelt Corollary to the Monroe Doctrine. This was Theodore Roosevelt's declaration that the US would step in whenever any country in the Western Hemisphere was showing that it couldn't really run itself properly. As the link below says, TR declared
that “chronic wrongdoing” or “impotence” on the part of neighboring countries might force the United States to exercise “an international police power,”...
These are the most important ways in which the US expanded its presence and influence in the Caribbean in the time you mention.