Few parishes celebrate baptisms during Advent. There’s not a particular prohibition against them. December tends to be busy, if you didn’t notice lately.
Church life operates at a slower pace after December 25th. If the clergy haven’t gone off the ranch to vacations and such, there might be a lot of time to initiate an infant. When might the best time be?
In my family, the adoption of the young miss was achieved on December 17th, so baptism followed a few Sundays later. January 6th, in fact. That day, the Theophany to the Orthodox, folds a number of events as Christ shows forth into the world: Magi, the Temple rabbis, John the Baptist and other witnesses at the River Jordan. In the West we split those observances onto a number of different days.
Sit in the purple chair and render judgment, or at least influence in your parish. Would you encourage baptisms on any of the observances of the Christmas season? Baptism of the Lord? Epiphany? Holy Family? I suppose a good case could be made for any of them.
Or do you live in a cold clime? Is it bad to trot babies out into the frigid air? Or more significantly, do parents feel this way?
Catholic Sensibility 
Correction, most Orthodox (Byzantine and Coptic) solely celebrate the baptism on Theophany. The gospel of Dec 25. is from the visit of the Magi with the Vigil being read from Luke. I believe that the Armenian Orthodox Church is the only body that celebrates the Nativity (with the visit of the Magi) and Baptism together on their January 6th calendar, which on our civil calendar is Jan. 19.
Good to know; thanks Devin