See below for links to Benvolio's character analysis, and a website which gives you every quote Benvolio ever says. If you want a single quote that sums up what Benvolio does in the play, I'd go for this one - actually the first thing he says in the play:
He beats down their swords.
Put up your swords. You know not what you do.
Benvolio, never in love or fighting himself, is constantly trying to persuade Mercutio and Romeo to chill out and stop being in love, or stop fighting, or trying to fight. And after Mercutio's death, he just vanishes from the play. He is the traditional peacemaker.
And, incidentally, his name means "Good-wishing" or "well-wishing" (Ben - "good", and volio - "I wish").
In Act three, scene 1, Benvolio says:
I pray thee, good Mercutio, let's retire.
The day is hot, the Capels abroad,
And if we meet we shall not 'scape a brawl,
For now, these hot days, is the mad blood stirring.
That passage typifies Benvolio, the level-headed peacemaker, who is speaking wisely to the hot-blooded Mercutio. While both Romeo and Mercutio tend to be dramatic and impulsive, Benvolio is the steady friend who tries to steer them from danger. This strategy on Benvolio's part has a history of not working. Early in the play, in Act one, scene 1, Benvolio advises the lovesick Romeo to seek out other women besides Rosaline, saying, essentially, that there are lots of attractive women in Verona:
Be ruled by me, forget to think of her ... Examine other beauties.
We see how that works out: the cure is worse than the initial disease, as Romeo ends up falling even more madly and fatally in love with Juliet than with Rosaline.
In the first passage quoted, Benvolio clearly has a presentiment of danger, foreshadowing the events that will soon leave Mercutio dead, but Mercutio will have none of his friend's caution. Both Romeo and Mercutio run on heedlessly and die, whereas Benvolio survives.