That the wise men did not arrive until Jesus was a child and no longer an infant is derived from several Scriptures, particularly in Matthew.
Matthew chapter 2: KJV
8 And he sent them to Bethlehem, and said, Go and search diligently for the young child; and when ye have found him, bring me word again, that I may come and worship him also.
9 When they had heard the king, they departed; and, lo, the star, which they saw in the east, went before them, till it came and stood over where the young child was.
11 And when they were come into the house, they saw the young child with Mary his mother, and fell down, and worshipped him: and when they had opened their treasures, they presented unto him gifts; gold, and frankincense, and myrrh.
13 And when they were departed, behold, the angel of the Lord appeareth to Joseph in a dream, saying, Arise, and take the young child and his mother, and flee into Egypt, and be thou there until I bring thee word: for Herod will seek the young child to destroy him.
14 When he arose, he took the young child and his mother by night, and departed into Egypt:
In these Scriptures Jesus is referred to as a child and not a babe. In the following scripture is where it is thought that Jesus was at least a year old and maybe as old as two when the Wise men arrived.
Mat 2:16 KJV
Then Herod, when he saw that he was mocked of the wise men, was exceeding wroth, and sent forth, and slew all the children that were in Bethlehem, and in all the coasts thereof, from two years old and under, according to the time which he had diligently enquired of the wise men.
Even though this is not definitive it is a pretty strong argument that Jesus was no longer a babe.