I agree with pohnpei397--I don't think the Prince should punish anyone. In fact, I'm not sure that in the Prince's last lines ("Go hence to have more talk of these sad things./ Some shall be pardoned, and some punished") he's referring to the type of punishment that he would implement.
To look at the situation more accurately, we first need to look at the Prince's role in the play. After the brawl between the Montagues and the Capulets that occurs at the opening of the play, the Prince warns, "If ever you disturb our streets again/ Your lives shall pay the forfeit of the peace." This threat, obviously, is one that the Prince is compelled to impose because of the out-of-control fighting between the two families. It's the only way, after so many years--and even generations--of fighting that he feels he can keep more blood from being shed. Again, this is the Prince's responsibility given his position in Verona.
At the end of the play, while Montague and the Capulets look upon the bodies of their dead children and learn of their relationship with each other, the Prince observes that "some shall be pardoned, some punished." I think the punishment here (specifically, for the Capulets, Montague, and maybe Friar Laurence) is the guilt that they will experience that accompanies their role in this tragedy. Montague and the Capulets, in perpetuating the feud, have driven their children to tragic ends. Friar Laurence, though his intentions were good, could have gone about things differently; as a result, Romeo and Juliet--and others--are dead.)
Given the Prince's threats in Act 1, readers might assume that the Prince is the type of official who uses the threat of punishment more as a deterrent than as anything else. Further, it's important to note that even though his Act 1 threat warned that anyone involved in more fighting would be killed, he only banished Romeo for killing Tybalt. (Though we understand this to be a show of mercy, Romeo sees banishment as more of a punishment, as it prevents him from being with Juliet.) Thus, by no means does the Prince take pleasure in punishing members of either family. As pohnpei397 notes, the deaths of Romeo and Juliet are, in and of themselves, punishment enough for everyone else involved in the situation.
Obviously, this is a question that's very debatable. Good luck in deciding for yourself!
I assume you are talking about the situation at the end of the play...
I do not think that the prince should punish anyone. All the people who did any actual killing are themselves dead so you do not have the issue of needing to punish a killer.
You could argue that the familes (and especially the Capulets) are at fault and should be punished. But they have already been punished severely by losing their daughter (and Tybalt, I suppose). Their punishment has been worse than any "crime" they may be guilty of.
You can make a case for punishing Friar Lawrence. He tried to get around the wishes of the two families and he did things that led to Romeo and Juliet dying. So I suppose you could punish him. However, he had the best of intentions (making peace between the two families) and so I do not really think he deserves it either.
I don't think the prince should punish anyone.ALL THE PEOPLE who did any actual killing ara themselves dead so you do not have the issue of needing to punish a killer.