In Jules Verne's "Around the Moon" (1870) they already need to carry an oxygen supply. However, there is only a brief moment of weightlessness as the vessel crosses the point where lunar and terran gravity cancel each other. This denote a latest date when weightlessness was not yet common knowledge.

Tsiolkovsky certainly knew weightlessness in the 1880s, and didn't deem it necessary to explain it very much. It seems safe to assume that it was accepted among experts.

It's hard to say when it became common knowledge. It's entirely possible that every physics teacher from 1870 onward found it worthwhile to point out the obvious mistake in Verne's story, but I wouldn't know where to look for a source. It's also hard to say whether a bit of trivia that doesn't affect anyone can constitutes "common" knowledge.

It certainly became more widely known when rocketry and the prospect of space flight became positively popular in the 1920s. When Fritz Lang made "The Woman in the Moon" in 1929, the depiction of weightlessness was no longer optional.

However, all of that is nothing compared to actual spaceflight becoming the topic of evening news. Between Sputnik and Mercury, truly everybody had learned about vacuum and weightlessness, whether they had even the slightest interest in the topic or not.