I think that the answer to this can be found in Act I, Scene 1. The answer is that it is a character named Antony, or Mark Antony.
But it is not really right to say that just touching this guy in general will allow a sterile woman to have a baby. If you just saw him on some regular day, he would not have this power. But on this day, he has this power. The reason is that this day is the Lupercalia. It is a fertility celebration.
In Rome on this day, men would run through the city carrying these little whips made of fur. Anyone they hit with one was supposed to have an easier time conceiving and bearing children.
Forget not, in your speed, Antonius,
To touch Calpurnia; for our elders say,
The barren, touched in this holy chase,
Shake off their sterile curse.
Act I Sc.2 of "Julius Caesar" takes place on February 15th. Traditionally it was the day on which the festival Lupercalia was celebrated in honor of Lupercus the god of the shepherds. A popular superstition was that if a person who takes part in the athletic race held on that day touched a barren woman she would conceive.
The stage direction at the beginning of the scene reads, "ANTONY, for the course," which means that Antony has come dressed to run in the race. Traditionally the persons who ran in the race during the festival of Lupercalia were dressed in goat skins. Caesar reminds Antony of this superstition and requests him in all the hustle and bustle of the celebrations and the race not to forget to touch Calpurnia. What is ironic is that Caesar, who was such a mighty emperor, should have faith in such a silly superstition.