This is a surprisingly complicated question. Let us look at both sides.
We know that the events of Genesis 37 took place twelve years before Isaac’s death. Since Joseph is described as “the son of his [Isaac’s] old age” (37:3), one might well think that, at that time, he was the youngest son, i.e., Benjamin was not yet born. This is given some credibility by the description of Benjamin as “a child of his [Isaac’s] old age, a little one” (44:20); at that time, Joseph will be 39, and so 22 years will have passed. If Benjamin was born before Gen 37 opens, then that would entail that when the whole family goes to Egypt, Benjamin would be at least 22 years old. It seems strange to refer to a grown man of 22 years as “a child...a little one”.
Perhaps attention to the Hebrew words would permit this, though. “Child” translates יֶלֶד or yeled, glossed “child, son, boy, youth”; while typically this means “child,” a significant amount of time it means “youth” or “young man” and thus can refer to a young grown man. And where the KJV has “a little one,” the Hebrew is simply קָטָן or qatan, glossed “small, young, unimportant”. Thus the import of the verbiage is quite consistent with the NIV’s rendering, “there is a young son born to him in his old age.”
Now, is there any evidence that he was born before the events of Gen 37? Actually, there is. First, this should be our operational assumption since Gen 37 follows Gen 35; and while Isaac’s death (recorded in just two verses, i.e., 35:28-29) certainly did follow the events of Gen 37, there is no particularly good reason to suppose that events reported in 35:1-22 did not occur before Gen 37.
Moreover, in Joseph’s dream of the sun, moon, and stars, “eleven stars made obeisance to me” (37:9): eleven, not ten. That suggests Benjamin had been born. The problem with this as evidence is that the text makes it perfectly clear that this dream was inspired by God, and thus only God needed to know the number of brothers who would bow down to Joseph. But it is, certainly, some evidence.
The same dream also provides evidence on the other side. Jacob asks, in response to this dream, “Shall I and thy mother and thy brethren indeed come to bow down ourselves to thee to the earth?” (37:10) The Hebrew for “thy mother” (וְאִמְּךָ֣ or way’immaka, “and your mother”) is in a singular form, i.e., not “mothers”; if he does not use the word “mothers,” this suggests only that only Rachel is meant. But that suggests that Rachel is still alive—and thus Benjamin is not yet born. Theoretically, however, Leah or Bilhah could have taken the place of Rachel as Joseph’s mother. Another possibility is that Jacob misunderstood the prophetic meaning of the sun and moon bowing down.
Another solid bit of evidence that Benjamin was born before the sale of Joseph is that Joseph was later insistent upon the brothers bringing their youngest brother to him (42:15-16)—suggesting he had a special interest in him, and thus, had known and thought about him for all the years he had been in Egypt.
On balance, it seems more likely that Benjamin was born before the events in Gen 37, but the evidence does not appear conclusive.