Old English sincan (intransitive) "become submerged, go under, subside" (past tense sanc, past participle suncen), from Proto-Germanic *senkwan (source also of Old Saxon sinkan, Old Norse sökkva, Middle Dutch sinken, Dutch zinken, Old High German sinkan, German sinken, Gothic sigqan), from PIE root *sengw- "to sink."
The transitive use (mid-13c.) supplanted Middle English sench (compare drink/drench) which died out 14c. Related: Sank; sunk; sinking. Sinking fund is from 1724. Adjective phrase sink or swim is from 1660s. To sink without a trace is World War I military jargon, translating German spurlos versenkt.