What is the biblical basis for commemorating the birth of Jesus? For many Christian denominations, Christmas is the highest holiday of the year.
What is the biblical basis for commemorating/celebrating the birth of Jesus, regardless of the particular day of the year on which it is celebrated?
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.