We are an economic and military superpower and have been for more than a century, so our "American lake" as Teddy Rosevelt called the Caribbean has been subject to a great deal of both political and economic influence.
Many people believe Puerto Rico may one day become the 51st state of the USA. They are currently a territory, but the large island's people still maintain their own culture and way of life. Like their territorial neighbors to the east, the U. S. Virgin Islands, they enjoy many of the perks and luxuries of America without completely giving up their West Indian identity.
Concerning how U.S.A. has had influence in Puerto Rico and Haiti: I grew up with a pack of hunting dogs. Most of them were kept tied, but a few always ran loose in the yard. For years, Leo, the patriarch of the whole bunch, ruled the yard. Ocassionally Leo would have to snap at another dog to maintain his dominance. Then Leo got old, and Boss got old enough and big enough and mean enough to take on Leo. Boss defeated Leo. Afterwards, Boss ruled the yard and he was so big that he did not even have to snap at the other dogs to maintain his dominance; they just naturally deferred to him.
Well... the United States has owned Puerto Rico ever since the Spanish American War back in 1898, so that would lead to it having a lot of influence.
And the US occupied Haiti for about 20 years from 1915 to 1934.