This is a great question to consider. I think personally that the character of Bassanio is a fascinating example of a character who occupies the social position of a gentleman whilst at the same time showing little evidence of acting as we feel a gentleman should act. We could therefore argue that this play, like other works of literature such as Great Expectations, helps us to analyse the difference between what we can term gentlemen and gentle-men, or men who are gentlemen through class and those who actually act like gentlemen, but may not occupy that position.
Personally, I have never been that impressed with Bassanio as a character, and fail to understand why Portia was so excited about him. Let us remember that in Act I scene 1, Bassanio uses his relationship with Antonio (that may or may not be based on homosexuality) to manipulate Antonio into giving him more money after he confesses how he has wasted his own fortunes:
'Tis not unknown to you, Antonio,
How much I have disabled mine estate
By something showing a more swelling port
Than my faint means would grant continuance.
The way in which he is very open about his profligate ways to the man who he knows loves him and will do anything for him shows that Bassanio is an irresponsible and calculating individual who looks for others to help him out of the trouble his own actions have landed him in rather than taking responsibility for his difficulties himself. The way he describes Portia and her wealth likewise shows us that his real aim is the mercenary possession of her wealth to fund his spendthrift ways. Therefore, at least of the beginning of the play, I don't think we can argue that Bassanio displays the character of a gentleman at all, even if he is, by birth, a gentleman.