Yes, people can fast on Mawlid to welcome the night of the Prophet's birthday and with the intention of worship if they want.
There is nothing in the Qur’aan to say that we should celebrate the Mawlid or birthday of the Prophet (peace and blessings of Allaah be upon him). The Prophet himself (peace and blessings of Allaah be upon him) did not do this or command anyone to do it, either during his lifetime or after his death. Indeed, he told them not to exaggerate about him as the Christians had exaggerated about Jesus (upon whom be peace). He said: “Do not exaggerate about me as the Christians exaggerated about the son of Maryam. I am only a slave, so say, ‘The slave of Allaah and His Messenger.’” (Reported by al-Bukhaari).
What has been reported is that the Prophet (peace and blessings of Allaah be upon him) made the day of his birth a day of worship, which is different to celebration. He was asked about fasting on Mondays, and he said: “That is the day on which I was born and the day on which I was entrusted with the Mission or when I was first given Revelation.” (Reported by Muslim, al-Nisaa’i and Abu Dawood).