Until Luke's Gospel was written, around the end of the first century CE, there was no suggestion that the year of John's birth was known to Christianity. This gospel introduced the account of the birth of John the Baptist, placing it around 6 months before the birth of Jesus. The same gospel is very ambiguous as to when Jesus was born. On the one hand, it says that Jesus was born during the reign of King Herod, who died in 4 BCE, but on the other hand during the census of Quirinius, which took place 6 CE, ten years after Herod's death. If Jesus was born during the reign of King Herod, then John the Baptist was born no later than 5 BCE, but probably several years earlier. If Jesus was born during the census of Quirinius, then John the Baptist was born in 5 CE or very early in 6 CE.
The majority of New Testament scholars dismiss the story of the birth of John the Baptist as unhistorical. Uta Ranke-Heinemann (Putting Away Childish Things) says that both Zacharius and Elizabeth were literary creations by the author of Luke's Gospel. So, on the scholarly view, we do not really know when John the Baptist was born.
The synoptic gospels say that John was executed in Galilee shortly after the start of the mission of Jesus, which is generally placed around 29 CE. However, the first-century Jewish historian Josephus says that John was executed for different reasons than given by the gospels, in a very different location and as late as 35 or 36 CE.