It is essential to respond to those who defame the Prophet (peace and blessings of Allaah be upon him)

Defaming the Prophet (peace and blessings of Allaah be
upon him) is a kind of kufr. If that is done by a Muslim then it is apostasy
on his part, and the authorities have to defend the cause of Allaah and His
Messenger (peace and blessings of Allaah be upon him) by executing the
one who defamed him. If the one who defamed him repents openly and is
sincere, that will benefit him before Allaah, although his repentance does
not waive the punishment for defaming the Prophet (peace and blessings
of Allaah be upon him), which is execution. 

If the person who defames him is a non-Muslim living under a
treaty with the Muslim state, then this is a violation of the treaty and he
must be executed, but that should be left to the authorities. If a Muslim
hears a Christian or anyone else defaming the Prophet (peace and
blessings of Allaah be upon him) he has to denounce him in strong terms. It
is permissible to insult that person because he is the one who started it.
How can we not stand up the Prophet (peace and blessings of Allaah be
upon him)? It is also obligatory to report him to the authorities who can
carry out the punishment on him. If there is no one who can carry out the
hadd punishment of Allaah and stand up for the Messenger (peace and
blessings of Allaah be upon him) then the Muslim has to do whatever he can,
so long as that will not lead to further mischief and harm against other
people. But if a Muslim hears a kaafir defaming the Prophet (peace and
blessings of Allaah be upon him) and he keeps quiet and does not respond for
fear that this person may then defame him even more, this is mistaken
thinking. With regard to the verse (interpretation of the meaning): 

“And insult not those
whom they (disbelievers) worship besides Allaah, lest they insult Allaah
wrongfully without knowledge”

[al-An’aam 6:108], 

this does not apply in cases where they defame Allaah and His
Messenger (peace and blessings of Allaah be upon him) first. Rather
what is meant is that it is forbidden to insult the gods of the mushrikeen
first, lest they insult Allaah out of ignorance and enmity on their part.
But if they insult Allaah and His Messenger (peace and blessings of
Allaah be upon him) first, then we must respond and punish them so as deter
them from their kufr and enmity. If we leave the kuffaar and atheists to say
whatever they want without denouncing it or punishing them, great mischief
will result, which is something that these kuffaar love. No attention should
be paid to the one who says that insulting or responding to insults will
make him more stubborn. The Muslim has to have a sense of protective
jealousy and get angry for the sake of Allaah and His Messenger (peace
and blessings of Allaah be upon him). Whoever hears the Prophet (peace
and blessings of Allaah be upon him) being insulted and does not feel any
protective jealousy or get angry is not a true believer – we seek refuge
with Allaah from humility, kufr and obeying the Shaytaan. 

And Allaah knows best.