The fame of Socrates and the way in which he is now perceived depends almost entirely on Plato. He would certainly be a minor figure if he appeared only in the accounts of Xenophon and as the object of ridicule in Aristophanes's. In his dependence on the spoken word, Socrates is also more akin to a religious leader than a philosopher, making the events of his life more of a focus than is the case with Aristotle or with Plato himself (who is often difficult to separate from the persona of Socrates).
Agreeing with the proposition, one might contend that the sensational result of Socrates's trial and his famous refusal to escape death in Plato's dialogue Crito are two of the most widely known stories about him. It is these events that make Socrates a popular hero who died for his beliefs, rather than just another of the many Greeks who talked about philosophy.
Disagreeing with the proposition, one might point out that almost all Plato's writing about Socrates, including the crucial role he plays in the Republic, takes place before he was sentenced to death. His achievement as a philosopher would barely have been affected at all. Moreover, no one can know how much longer Socrates would have lived and what else he might have said. There might have been twenty more Plato dialogues featuring him after his release from custody, each one featuring brilliant new insights. He might even have written a book himself.
One way to approach this question might be to make a distinction between Socrates as a figure in history and as a philosopher. His importance might be enhanced by the manner of his death in the former role but not in the latter.