How did the US expand its influence and presence in the Caribbean up to 1917?

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

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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.