Calpurnia is Caesar’s wife. Her primary function is to show weakness in the character of Caesar, and in most parts of the play, up until Caesar’s death, that’s about all she does. After Caesar’s death she disappears and thus has no continuing function in the play.
The fact that Calpurnia is barren helps characterize Caesar. One of Caesar’s goals is to have a child, and he makes this desire known when he asks Antony to touch her during a race at a holy festival. The superstition is that when a woman is touched in this way she is no longer barren. The fact that Caesar’s wife is barren implies a weakness about Caesar, who is mighty militarily and in the eyes of many Romans.
Calpurnia’s character also serves to show that Caesar, although he would never admit it openly, is actually afraid of the conspirators. This happens when Calpurnia begs Caesar not to go to the Senate. At first Caesar says he is not afraid and will not be seen as such. After a few more words from Calpurnia, however, Caesar abruptly changes his mind and says he will heed her advice and stay at home. It is as though he were waiting for a reason to avoid the danger that the conspirators represent.
Of course, Caesar ends up going anyway after being flattered by Decius, and is killed.