1824 of shoes, 1839 of trousers, 1920 of gowns, 1931 of brassieres, from strap (n.) + -less.
Entries linking to strapless
strap n.
1610s, "band of leather," from Scottish and/or nautical variant of strope "loop or strap on a harness" (mid-14c.), probably from Old French estrop "strap," from Latin stroppus "strap, band," perhaps via Etruscan, ultimately from Greek strophos "twisted band; a cord, rope," from strephein "to turn" (from PIE root *streb(h)- "to wind, turn"). Old English stropp, Dutch strop "halter" also are borrowed from Latin, and the Old English word might be the source of the modern one. Slang meaning "credit" is from 1828.
-less
word-forming element meaning "lacking, cannot be, does not," from Old English -leas, from leas "free (from), devoid (of), false, feigned," from Proto-Germanic *lausaz (cognates: Dutch -loos, German -los "-less," Old Norse lauss "loose, free, vacant, dissolute," Middle Dutch los, German los "loose, free," Gothic laus "empty, vain"), from PIE root *leu- "to loosen, divide, cut apart." Related to loose and lease.