MOST ENGLISH-SPEAKING Caribbean democracies, or would-be democracies, are in a state of explosive economic crisis, and their governments are increasingly taking it out on the press. In Grenada, the uncertain new "People's Revolutionary Government" of Maurice Bishop has just closed the Torchlight, the only independent paper, complaining about, among other things, the paper's accounts of Mr. Bishop's growing ties with Cuba: he received a shipload of Cuban arms 12 hours after he put in his request. In Guyana, Prime Minister Forbes Burnham, in a heavy sea, has cut off the Mirror's access to newsprint. Again, the Mirror is the only independent voice.

In Jamaica, the situation of the Gleaner newspapers, also the only non-government voice, is parlous. While in the opposition, Michael Manley hailed the Daily Gleaner as "a truly great newspaper." Since becoming prime minister in 1972, he has conducted an escalating battle with the paper, which has vigorously challenged his links with Cuba and the domestic policies that have contributed, in arguable measure, to the unarguable devastation of the Jamaican economy. After the Cuban ambassador denounced the Gleaner last month and the paper responded by demanding the envoy's ouster, Mr. Manley personally took part in a demonstration at the newspaper's offices. Ominous threats against it have been made.

Hemispheric press groups have spoken out for these embattled papers, The State Department has, too, and its concern is especially welcome. Early on, the Carter administration set out to support left-leaning Third World democracies -- if they served their people. Jamaica benefitted handsomely. Mr. Manley, however, has lost much of the sympathy he earlier enjoyed. Publicly he is given a friendly face, but privately some officials lament that he is running his country into the ground. His rhetoric (Puerto Rican independence advocates must be supported "regardless of whether they are in the minority at this time") has not helped. Jamaica does not appear to be among those tippy countries to which Mr. Carter promised extra aid in his Oct. 1 speech on communism and the Caribbean situation.

Jamaica is one small country among many in the region, and its affairs are not always followed closely by either the American press or the American bureaucracy. But nothing would send Americans a clearer signal of Mr. Manley's abandonment of democratic standards than to have the Gleaner suffer the fate of the Torclight and the Mirror.