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.