It is not prescribed to celebrate the birthday of anyone, whether Prophets or anyone else, because that is not narrated in shareeâah. Rather it is something that has been taken from non-Muslims, such as the Jews, Christians and others.
See the answer to question number 10070 and 13810.
What is meant by celebrating birthdays here is celebrating on the day on which a person was born, such as celebrating the 12th of Rabeeâ al-Awwal which some people believe is the day on which the Prophet (blessings and peace of Allah be upon him) was born.
With regard to speaking about the Prophet (blessings and peace of Allah be upon him) and teaching about him, mentioning his good qualities, virtues and Sunnahs, this is mustahabb and is prescribed at all times, and this is not called a Mawlid, just as celebrating a wedding is not called a Mawlid, but it is common in some Muslim countries to call every celebration that is done in an Islamically acceptable manner, with no dancing, music or mixing, a Mawlid, and they say: we will do a Mawlid on the wedding day or on the circumcision day, and a preacher comes to exhort the people, and a reader comes to read Qur'aan, and so on. There is no basis for giving it this name, and calling it by this name does not change the ruling concerning it. There is nothing wrong with people celebrating the wedding and having someone to address the people and exhort them and remind them of good, or speak about the Prophet (blessings and peace of Allah be upon him) and mention his biography and good characteristics. This is prescribed in Islam, and does not come under the heading of celebrating the innovated Mawlid.
There is nothing wrong with holding activities or meetings in the mosque to teach people about the Prophet (blessings and peace of Allah be upon him), without singling out a particular day because of some belief in its virtue, such as the day of the Mawlid or the 15th of Shaâbaan (al-nusf min Shaâbaan) or the day of the Israâ and Miâraaj. Rather that should be done on any ordinary day. There is also nothing wrong with offering food to the people who attend, but it is important to publicise the ruling that this should not be called a Mawlid, and it does not come under the ruling on celebrating the Mawlid, so that no one will think that celebrating the Mawlid is prescribed in Islam.
We ask Allah to help you to act in accordance with the Sunnah of the Prophet (blessings and peace of Allah be upon him) and to propagate it among people.
And Allah knows best.