What is the biblical basis for commemorating the birth of Jesus?

Christmas isn't a birthday party. It is a feast day for the incarnation of the logos as mentioned in:

John 1:14 NIV

The Word became flesh and made his dwelling among us. We have seen his glory, the glory of the one and only Son, who came from the Father, full of grace and truth.

It could be celebrated on any day. The reasons the day in December was chosen is well documented.

Just to reiterate, it is not a celebration of a birth. It is the celebration for the incarnation which happened to be a birth. The celebration of a birth is not biblical. Furthermore, if you're Catholic:

The Catholic Encyclopedia, 1913, Vol. X, p. 709

“Origen [a writer of the third century C.E.] . . . insists that ‘of all the holy people in the Scriptures, no one is recorded to have kept a feast or held a great banquet on his birthday. It is only sinners (like Pharaoh and Herod) who make great rejoicings over the day on which they were born into this world below.’”

I do not think there is a Biblical basis for celebrating Nativity per se, however there is are several Biblical bases for observing the traditions that the Apostles and their successors passed down (e.g. 2 Thessalonians 2:15). The mandate to observe the Feast of the Nativity is found in the Apostolic Constitutions, Book V, Section III.XIII:

Brethren, observe the festival days; and first of all the birthday which you are to celebrate on the twenty-fifth of the ninth month; after which let the Epiphany be to you the most honoured, in which the Lord made to you a display of His own Godhead, and let it take place on the sixth of the tenth month

(Note that the "first" month in antiquity was what we now call April, not January)

The biblical basis is that the angel told the shepherds to celebrate the first Christmas, and the angels themselves were celebrating it:

But the angel said to them, “Do not be afraid; for behold, I bring you good news of great joy which will be for all the people; for today in the city of David there has been born for you a Savior, who is Christ the Lord. This will be a sign for you: you will find a baby wrapped in cloths and lying in a manger.” And suddenly there appeared with the angel a multitude of the heavenly host praising God and saying,

“Glory to God in the highest, and on earth peace among men with whom He is pleased.”

When the angels had gone away from them into heaven, the shepherds began saying to one another, “Let us go straight to Bethlehem then, and see this thing that has happened which the Lord has made known to us.” So they came in a hurry and found their way to Mary and Joseph, and the baby as He lay in the manger. When they had seen this, they made known the statement which had been told them about this Child. And all who heard it wondered at the things which were told them by the shepherds. But Mary treasured all these things, pondering them in her heart. The shepherds went back, glorifying and praising God for all that they had heard and seen, just as had been told them. (Luke 2:10-20. NASB)

Which, importantly, shows the Christian way of celebrating Christmas: seeking Jesus, contemplating Him, acknowledging Him as your Teacher, Savior, Lord and God ("An ox knows its owner, and a donkey its master's manger, but Israel does not know, my people do not understand." Isaiah 1:3), opening your mind and heart to Him to let his word enlighten and transform you, receiving Him in Holy Communion (Bethlehem = "House of Bread"), thanking and glorifying God the Father for giving his Son to us, and then telling others "what was from the beginning, what we have heard, what we have seen with our eyes, what we have looked at and touched with our hands, concerning the Word of Life" (1 John 1:1).

No room for "Santa", for gifts, for elaborate meals, for decoration, all of which is a waste of time and energy which distracts us from benefiting from the invaluable Gift that God the Father gave to us: his Consubstantial Son.

John's Gospel Chapter 1 tells us that in the beginning was the Word, and the Word was with God, and the Word was God. (v1) and that this Word became flesh and dwelt among us (v14). So Christians beleve the Birth of Christ entailed the Incarnation (becoming flesh) of God Himself as a human person. This, if true, was one of the most important things ever to happen.

Psalm 145 v4 says "one generation shall praise thy works unto another, and shall declare thy mighty acts".

Many Christians feel that the Birth of Christ, the Incarnation, is a mighty act which should be declared to future generations.

There are many ways this could be done, some Biblically-based, some not. In the Book of Exodus God instituted the Jewish festival of Passover. Chapter 12, verses 26 and 27 may explain why: "It shall come to pass, when your children shall say unto you What do you mean by this service? That you shall say it is the sacrifice of the Lord's passover, who passed over the houses of the children of Israel in Egypt." God instituted the Passover festival so that future generations should learn what had happened.

In the book of Esther the Jews were delivered from an impending calamity. To commemorate this the annual Jewish festival of Purim was established.Chapter 9 verse 27 tells us the Jews "took it on themselves" to establish that they and their descendants should celebrate this annually for ever. God did not institute Purim, as He instituted Passover, Rather the Jews took it on themselves to do so.

The Bible does not command the celebration of Christmas, any more than Purim, but the Church took it on itself to do so.A Biblical basis is that the event it commemorates is important enough to be passed down to future generations, that God set the example with Passover of using an annual festival to pass things down, and that the Jews had also later taken it on themselves to establish an annual festival without God's direct command, as a way to perpetuate knowledge of something they felt important. This is the Biblical way, so this is what the Church did.

If looking for a Biblical basis for the date of Christmas, it is important to bear in mind there may not be one. Elizabeth, John the Baptists mother, was already 6 months pregnant when Gabriel announced to Mary that she too would bear a Child (Luke 1 v26). This puts the Birth of Christ around 15 months after the conception of John. Gabriel announced that Elizabeth would conceive when he appeared to John's father, Zachariah, when he was burning incense in the temple (Luke 1, vv10 11) . There was a tradition that this happened on the day of Yom Kippur, (late Sptember/early October). We are also told that Zachariah was of the course of Abijah (Luke 1 v5) - this indicates his place on the priestly roster and thus the weeks in the year when he would have been on duty. Attempts have beeen made to determine when this was, but there is no consensus.