Bassanio finally makes his choice between the caskets in Act III scene 2, and it is important to recognise how Shakespeare increases the audience's anticipation by showing two other hopeful suitors going through exactly the same process that Bassanio is about to go through and failing. This of course increases the dramatic irony of the test: the audience, just like Portia, knows which casket Bassanio needs to pick in order to be successful.
Bassanio's thinking process is revealed to us in his soliloquy as he muses over the caskets. His thinking begins with the thought that "The world is still deceiv'd with ornament," and he goes on to cite a number of examples. In law for example pleas that are made with "gracious voice" often obscure their true evil intent. In the same way in religion "damned errors" are blessed and covered up. Men ostensibly appear to be as brave as Hercules but in fact, underneath this tough exterior, are cowards. Bassanio comments that there seems to be an inverse relationship between appearance and reality: what something appears to be is often precisely the opposite compared to the reality:
Thus ornament is but the guilded shore
To a most dangerous sea: the beauteous scarf
Veiling an Indian beauty; in a word,
The seeming truth which cunning times put on
To entrap the wisest.
On this basis, Bassanio rejects both gold and silver, opting for lead, because it offers no promises and because its "paleness" moves Bassanio more than eloquence. He is reluctant to trust in appearances and so opts for the casket that seems to offer least and in fact delivers all.