The quote, "Down with the Capulets!  Down with the Montagues!" occurs in Act I, Scene 1.  The quote is said by the citizens of Verona because they have again caused a civil brawl.  This says that they should be brought to justice because they are constantly disturbing the peace and ruining the streets and the citizens' businesses.  They upset their market place and ruin their livelihood with their constant fighting.  The citizens are sick of the brawls and want justice to be done.

This line is connected to the officers of the watch who are charged with keeping the peace. Also, as there is currently fighting in the streets, the watch would have the task of stopping the fight, preferably before the prince arrives.

This makes more sense within the full context of the citizen's lines:

Clubs, bills, and partisans! strike! beat them down!
Down with the Capulets! Down with the Montagues!

The line just prior to the one you've asked about seems to be a call to arms ("clubs", "bills" and "partisans" are all weapons), and the order to "strike" and "beat them down" suggests that the citizen is calling for weapons to help "beat down" the weapons from both sides. Hence, "down with the Capulets" and "down with the Montagues" might be read as a call to quiet (or put down) the weapons and the fighting forces on both sides.

This is said by a citizen of Verona in Act I, Scene 1. A fight has broken out in the streets of Verona between the servants of the Capulets and the Montagues. This citizen is obviously outraged that the feud between the two families has caused yet another fight, and he blames both of the families.