We don't have any surveys to determine what the general population believed. But it does seem clear that for the last 2000 years the vast majority of the educated population knew that the Earth was round.
... Wait, hold on. Is it not true that many who opposed Marco Polo's voyage did so on the basis that he would fall off the edge of the world?
... Because, y'know, it is...
No one opposed Marco Polo when he left. In fact, he went to China at age 6 with his father and uncle. Later he went back. But when he came back to Europe, many thought he made up his adventures.
If you are thinking of Christopher Columbus, he was also not opposed based on 'flatness'. This is a myth that came about quite recently:
No one before the 1830s believed that medieval people thought that the earth was flat.
The idea was established, almost contemporaneously, by a Frenchman and an American, between whom I have not been able to establish a connection, though they were both in Paris at the same time. One was Antoine-Jean Letronne (1787-1848), an academic of strong antireligious prejudices who had studied both geography and patristics and who cleverly drew upon both to misrepresent the church fathers and their medieval successors as believing in a flat earth, in his On the Cosmographical Ideas of the Church Fathers (1834). The American was no other than our beloved storyteller Washington Irving (1783-1859), who loved to write historical fiction under the guise of history. His misrepresentations of the history of early New York City and of the life of Washington were topped by his history of Christopher Columbus (1828). It was he who invented the indelible picture of the young Columbus, a "simple mariner," appearing before a dark crowd of benighted inquisitors and hooded theologians at a council of Salamanca, all of whom believed, according to Irving, that the earth was flat like a plate. Well, yes, there was a meeting at Salamanca in 1491, but Irving's version of it, to quote a distinguished modern historian of Columbus, was "pure moonshine. Washington Irving, scenting his opportunity for a picturesque and moving scene," created a fictitious account of this "nonexistent university council" and "let his imagination go completely...the whole story is misleading and mischievous nonsense."