Shakespeare never reveals why Antonio is so melancholy--even from the play's beginning. Personally, I believe that his depression results from loneliness and from his realizing that he has spent his life making money and belittling his competition (Shylock) and has nothing to show for it. Even though Antonio seems to be well-respected and considers Bassanio a friend, the audience must wonder if he has any true friends. Bassanio is always asking something from Antonio, and the only time that he tries to help Antonio is when Antonio is put in the dangerous position of forfeiting a pound of flesh (a position that he is in because of his own foolishness and because of Bassanio's borrowing money from him).
Similarly, at the end of the play when Antonio's suggestion for Shylock is upheld and all the couples are reunited, Antonio has no one. Granted, he still has his wealth, but he has no one to share it with, not even a friend. I think that Antonio realizes that he is not unlike Shylock, someone whom he constantly berated and disdained. For, like Shylock, Antonio is wifeless, childless, and perhaps even a little faithless after the trial.
I do not know that we are told that Antonio never smiles -- it's someone else that Portia says that about. But we are told that Antonio is always sad. In fact, those are the first lines of the play:
In sooth, I know not why I am so sad:
It wearies me; you say it wearies you;
But how I caught it, found it, or came by it,
What stuff 'tis made of, whereof it is born, 5
I am to learn;
And such a want-wit sadness makes of me,
That I have much ado to know myself.
As you can see from these lines, Antonio himself does not know why he is sad -- Shakespeare does not choose to tell us. We know he is not worried about his money and he is not worried about love. So what is he sad about? Some people who analyze the play say he is sad because he has chosen to be a merchant, making money off of other people, even though he does not like to use other people (at least not other Christians) for his own gain.