drape has 7 distinct synonyms spread across 2 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.
arrange in a particular way
“drape a cloth”
place casually
“The cat draped herself on the sofa”
cover or dress loosely with cloth
“drape the statue with a sheet”
hanging cloth used as a blind (especially for a window)
the manner in which fabric hangs or falls
“she adjusted the drape of her skirt”
a sterile covering arranged over a patient's body during a medical examination or during surgery in order to reduce the possibility of contamination
Synonyms, antonyms and the broader and narrower relations for drape come from Princeton University’s WordNet. Full sources and licences.