Cricket is indeed played on most Caribbean islands due to the large British influence most smaller islands had. Many slaves adopted the sport from their owners, and/or were encouraged to play it. By the 1900s it was the most popular sport in the Caribbean.
However, at about that time Baseball had become big in the U.S. and that popularity then spilled over to its immediate neighbors -- often intentionally. In 1916 for example U.S. forces occupied the Dominican Republic. The D.R. was experiencing a strong influx of Afro-Caribbean immigrants at that time, many of which brought with them Cricket. Most of these settled around the sugar cane growing area of San Pedro de Macoris.
Since the American occupying forces were not interested in seeing anyone play Cricket, they staged Baseball games instead, often bribing good Cricket players to switch over to Baseball. The ploy worked. Baseball took over as the leading sport on the island -- at least on the Dominican side. In Haiti they still play Cricket and especially Soccer.
It is interesting to note that many Dominican Baseball players in the Major League are from San Pedro de Macoris, which used to be the island's Cricket central.