I've seen this in various forms:
With pardonable exaggeration it was said that “A maiden bearing a nugget of gold on her head could wander safely throughout the realm.” (Morton & Lewis China: Its History and Culture)
Yet, as a direct consequence of Mongol tyranny, the Silk Road became safe for commerce, so safe that one traveler claimed “a young woman would have been able to travel with a golden tray on her head with no fear.” (Bergreen Marco Polo: From Venice to Xanadu)
It was said that during this time a virgin could cross the length of the Mongol Empire with a pot of gold on her head and never be molested. (Weatherford "Genghis Khan: Law and order")
But how common was this saying? What some attested instances of this being said?
The original uote comes from Juvaini's History of the World Conqueror and goes thus "For fear of his yasa and punishment his followers were so well disciplined that during his reign no traveller, so long as he was near his army, had need of guard or patrol on any stretch of road ; and, as is said by way of hyperbole, a woman with a golden vessel on her head might walk alone without fear or dread." The subsequent prevalence of this quote I can't really comment on though I would speculate it's popularity amongst current historians stems from 1) Juvaini's text is one of the most important contemporary works we have on the Mongols and 2) it serves as a nice way to illustrate the concept of the Pax Mongolica.
Sources:
Juvaini A A (n.d.) The History of the World Conqueror. Translated from Persian by Boyle J A (1958) Manchester: Manchester University Press, p. 272
Available online here: https://archive.org/stream/historyoftheworl011691mbp/historyoftheworl011691mbp_djvu.txt