Green’s Dictionary of Slang

(US Und.) a beggar’s crutches, a cane.

1913		Fort Wayne Jrnl Gaz. (IN) 19 Nov. n.p.: ’I’ll be goin’ out of here on sympathy sticks,’ meaning, of course, crutches.
1938		El Paso Times (TX) 16 Oct. 46/1: I have carried the sympathy sticks and been a wing ding artist.
1949		Monteleone Criminal Sl. (rev. edn).
1965		Beckly Post-Register (WV) 19 June 4/2: Crutches — ‘sympathy sticks,’ the doctor called them.
1998		Ottowa Citizen (Ontario) 26 July 12/3: She reaches for her cane - my ‘sympathy stick though it’s never brought me any sympathy’.