What biblical basis is there for written records being kept around Jesus's time

Without repeating what others have said, I note that you've asked about writing or recording.

In the ancient world (and quite unlike today), a lot of attention was paid to memorization. Not only were people expected to memorize, but there were specific techniques for memorizing large amounts of information, techniques which today have been all but forgotten. (You can learn more about this by reading Moonwalking with Einstein, a recent book about memorization techniques and history.)

Those ancient memorization techniques are strongly tied to a sense of place, or of spatial relationships. Things are better remembered if those things are tied to certain locations, as the human brain is optimized for remembering locations. With that in mind, read through the Gospel of Mark (for instance), and note how often the author calls attention to geographic locations in the text. Here are some examples from the first ten or so chapters of Mark:

• And as He walked by the Sea of Galilee...
• Then they went into Capernaum...
• Then He went out again by the sea...
• Now it happened that He went through the grainfields on the Sabbath...
• And He entered the synagogue again...
• And He went up on the mountain...
• Then they came to the other side of the sea, to the country of the Gadarenes.
• Now when Jesus had crossed over again by boat to the other side, a great multitude gathered to Him; and He was by the sea.
• Then He went out from there and came to His own country...
• When they had crossed over, they came to the land of Gennesaret and anchored there.
• From there He arose and went to the region of Tyre and Sidon.
• Again, departing from the region of Tyre and Sidon, He came through the midst of the region of Decapolis to the Sea of Galilee.
• Then He came to Bethsaida...
• Now Jesus and His disciples went out to the towns of Caesarea Philippi...
• Then they departed from there and passed through Galilee...
• Then He came to Capernaum.
• Then He arose from there and came to the region of Judea by the other side of the Jordan.
• Now as He was going out on the road...

Why did the author of the Gospel of Mark give such attention to place? I would put forth that this was probably a significant part of his, or his sources', technique for remembering parables that were told and things that took place.

The first thing that comes to mind is the Dead Sea Scrolls, which were hidden away before the end of the first century AD. The basic technology for writing was definitely present, as the scrolls were written on parchment and papyrus, with two different kinds of ink. Inkwells have been found among the ruins of the Qumran settlement that the Dead Sea Scrolls are connected with. So there's really no question that the notion of writing on paper-like materials with ink and some pen-like implement was familiar to the Jews of Jesus's day.

In Luke 4, Jesus read from the scroll of the prophet Isaiah. Jesus had not received the training of a priest or Pharisee. He was the son of a carpenter. So, it seems that this would indicate that literacy was not restricted to the ruling class of the priestly class.

Jesus challenged the Pharisees at one point, saying, "Have you never read...?"

Jesus may also have been writing actual words in the dirt in John 8 and the woman caught in adultery, though this is admittedly speculation.

Paul also indicates that he wrote letters with his own hand. Of course he had been a Pharisee.

In older times, Moses commanded the people in Deuteronomy 6 to observe the law and to write it on the door frames of your houses. This seems to assume the ability to write.

The king of Israel was commanded to write for himself a copy of the law and read it all the days of his life.

I could go on, but it seems there is significant evidence to support the idea that literacy was not uncommon among the Jewish people at the time of Christ.