Middle English sinken, from 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 "force or drag gradually downward" (attested from late 12c.) supplanted Middle English sench (compare drink/drench) which died out 14c. The sense of "go in, penetrate" (of a blow, a weapon, etc.) is from c. 1300; by early 14c. as "make a penetrating impression on the mind." Related: Sank; sunk; sinking.
From early 14c. as "be reduced to a lower or worse state;" late 14c. as "fall or fail as from weakness or under a heavy blow." From 1590s as "decrease, be reduced in value, amount, etc." Of the sun, moon, etc., "to set," c. 1600. Of land, "dip downward gradually," by 1726.
Contrasted with swim (v.) since 14c.; the adjectival phrase sink or swim is from 1660s. To sink without a trace is World War I military jargon, translating German spurlos versenkt.