How can we say that Antonio is a rich merchant?

There are many indications throughout the book that Antonio is a rich merchant.
First, Bassanio comes to Antonio to borrow some money so that he can travel to Belmont as a man of means. We do not know the exact equivalent of 3000 ducats today, but the money would have been a large sum. The amount is enough to help Bassanio stand out as a man of means when he visits Portia, who is a wealthy heiress. Further, the fact that it had been a habit of Bassanio's to borrow money from Antonio indicates that Antonio is a wealthy man. Apparently, Antonio has a habit of lending money to his friend but does not impose strict terms as Shylock does.
Another factor that shows Antonio as a wealthy merchant is that Shylock is willing to lend him the money. Shylock's primary goal is to make a profit from lending money. Although he places a demand for Antonio's flesh if the money is not repaid in time, his primary interest is to get the money paid back. Shylock would not have given the money to Antonio if he was not a reputable merchant with means to repay the money.
Lastly, Antonio owns a fleet of ships that travel across the world, bringing him revenue. Antonio is unable to extend the loan to Bassanio only because his assets have been held up at sea. The fact that he owns several ships with assets at sea is a clear indication that Antonio is a rich merchant.

Antonio certainly gives the appearance of being a wealthy man. He's a successful merchant with a large fleet of ships engaging in profitable trade throughout the world. He clearly has money to spare, as he often helps people out financially when they've fallen on hard times. People come to Antonio when they've got into debt with Shylock and can't pay him back. This is just one of many reasons why Shylock hates him so much.
In the past, Antonio has often bailed out his friend Bassanio, who always seems to be in need of money. But there are limits to Antonio's largesse. Because when Bassanio comes looking for another handout—this time because he wants to marry Portia—Antonio is forced to borrow the money from Shylock, which he must pay back on forfeit of losing a pound of flesh. It would seem that Antonio's generosity has finally caught up with him. But he wouldn't have been able to be so generous in the past had he not been so wealthy.