The language of the Silk Road in the medieval and post-medieval period, that is, from about 900 AD to 1850 was Persian (now often called Farsi).

This was the primary spoken and written language in the central portion of the road's transit, from Samarqand and the Ferghana valley (modern Uzbekistan) almost to Tabriz in Iran, and was understood in Turkistan (now Xinjiang, China) and well into what is now Turkey. It was also the native language of perhaps a majority of the road's merchants, whatever their ethnic origin might have been. Indian merchants all spoke Persian; so did Armenians. Persian was the literary language of Turko-Mongol governments that ruled the Silk Road territories from the 12th century as well. It was the perfect lingua franca.

Turkic dialects were of course understood, as they covered a similar range, but they were less rooted in urban areas and the sedentary population.

It's worth remembering that actual transit from China to Istanbul more or less stopped in the early 1500s as Iranian silk was then shipped directly west and Safavid Iran effectively closed its eastern borders to trade. The Silk Road then became two separate roads: one going from Iran to Aleppo or to Istanbul, and one, much less travelled, overland from China north of the Caspian Sea through Russia. In any case the maritime trade had become much more important.

James Corcoran, Religions of the Silk Road, Richard Frye, the Heritage of Central Asia, Janet Abu-Lughod, Before European Hegemony, any of the Cambridge Histories of Central Asia or of Iran or even Marco Polo's account discuss this pretty clearly. I don't have a complete bibliography handy so maybe someone else can dig up more sources.

EDIT: Adjusted a date.

Also it is good to remember "silk road" was not a single intact highway type of an entity but rather, a series of more or less connected trade routes. Ancient History doesn't know of a single person who would have travelled it from one end to another.