If Faramir had gone, then from a writer's point of view, Tolkien would have to find some other catalyst to get Frodo to leave.
Of course, I'm oversimplifying the plot. Tolkien (as genius as he was) could have given Frodo a million different reasons to leave...
But, through the character of Boromir so much is linked. Gondor, Aragorn, the White City, Boromir's frustration of being the one always having to defend Middle Earth (with only Rohan to help). Again, that narrow mindedness on Boromir's part that only Gondor defended Middle Earth just wasn't true. But to Boromir it was. That's why he wanted the ring. Gondor was failing, was falling... and there wasn't enough men left to defend it. So, he needed something stronger than the enemy to defeat the enemy. The ring was perfect. It was his only hope.
BOROMIR:
It is a gift. A gift to the foes of Mordor! Why not use this ring? Long has my father, the Steward of Gondor, kept the forces of Mordor at bay, by the blood of our people-- all your lands kept safe! Give Gondor the weapon of the enemy. Let us use it against him!
Also, when we meet these characters at the first, we can't telegraph the end on them. Meaning... when I am teaching during my acting classes, my actors may be on page 8 or 9 of the script and they begin to telegraph or react to what happens to their characters on page 32 or 33, I have to remind them that on page 8 your character doesn't have a clue that page 32 will happen to him. So, you have to play it as if you don't know.
So, to Boromir it was a weapon... it wasn't until he was appalled at his behavior concerning Frodo that he realized that Gandalf was right.
In Rivendell, Boromir was a good man who was desperate. At the end, at his hour of testing, he remained a good man. Well done, Boromir!