The carribean is an extremly diversified environment. It is not enough to get the english away from there. Still do the spanish own the largest island and half of the second largest. The netherland do own only small island, but they wwere quite important for the slave trade. Even the danish had one or two islands. I think it is impossible for one nation to control the carribean, if other don´t want them to. In OTL the spanish had it all, but it was very easy for the english and french to conquer the one or the other island (Jamaika was sacked rather accidently). The possesion of the smaller islands changed quite often.
To get one single nation to control the carribean, this aerea must become completly unintresting for any other nation. This is very difficult to construct, due to the centrality of the aerea in question.