desperate has 4 distinct synonyms spread across 4 of its 7 senses. Each sense below keeps its own list, because a synonym belongs to a meaning and not to a word: swapping one sense's substitute into another sense's sentence is how a thesaurus makes writing worse. WordNet records no antonym for it — antonymy is a relation between word forms rather than concepts, so it is deliberately sparse.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
arising from or marked by despair or loss of hope
“a despairing view of the world situation”
showing extreme courage; especially of actions courageously undertaken in desperation as a last resort
“made a last desperate attempt to reach the climber”
a person who is frightened and in need of help
“they prey on the hopes of the desperate”
2 of the 7 recorded senses carry no synonym, antonym or related term, so they are not shown above. Every sense, with its definition and examples, is on the full entry for desperate.
Synonyms, antonyms and the broader and narrower relations for desperate come from Princeton University’s WordNet. Full sources and licences.