In ancient Roman beliefs, Janus is the god of beginnings, transitions, and endings. He is depicted with two faces, one cheerful and one melancholy, symbolizing the uncertainty of the future. 

Solanio suggests that Antonio is as strange a figure as Janus, who is dual in his nature.

Nature hath framed strange fellows in her time:
Some that will evermore peep through their eyes
And laugh like parrots at a bagpiper,
And other of such vinegar aspect. (1.1.52-54)

Antonio insists that he does not know why he feels the sadness that he does, and he insists that he is not anxious about his merchant ships as his friends suggest. When another set of friends arrive, among whom is Bassanio, they, too, are concerned about Antonio's admitted sadness. He tells these friends: 

I hold the world but as the world Gratiano—
A stage, where every man must play a part,
And mine a sad one. (1.1.77-79)

Perhaps, Antonio is one of those men of the sea who seem to sense changes, such as storms and misfortune, before others. Antionio may be of such a temperament that he feels an inexplicable sadness at the time—a premonition, perhaps. Then, later on, his emotion becomes relevant. Certainly, that he is compared to Janus is significant because Antonio, too, looks in two directions: out to sea with his cargo and on land where he becomes involved with his friend Bassanio. So, his melancholy may foreshadow the worries to come for Antonio.