Jesus was Middle Eastern and Jewish. I doubt that He was very light skinned, although He could have been a ginger (ruddy) like King David.
Making Him white in paintings was the artists making Jesus look like themselves and is understandable.
He was depicted as black in the early Ethiopian church.
But if we want to go with a discussion I was part of decades ago, here's how my part went. As always I go to the Bible first. I just don't see much there there for tagging the Jews or Jesus with a skin color.
Jacob (Israel) had a twin brother named Esau (which means hairy) because he was covered in red hair (Genesis 25:25). Jacob's descendants are the Jews and Esau's the Edomites. I'm not sure if that means the Jews and Edomites looked like ginger haired Irishmen. There are a few traces of blacks with red hair. Just throwing that out there.
Then Jeremiah 13:22-24 has God, through Jeremiah, pointing out that the Jews can't help but keep on sinning, like a leopard can't change his spots, or an Ethiopian can't change his skin. That doesn't explicitly say that the Jews were a different color from the black Ethiopians, but it's at least suggestive that they were.
And Song of Songs 1:5-6 has the Shulamite woman (a Jew) say that she's deeply tanned, which could be a reference that she's a darker race than her lover. But I don't think so since the next verse alludes to her working out in the sun a lot (like a poor woman, who is grateful to be in love with a rich man who doesn't have to be out in the sun a lot). IMHO that could work for Jews being dark or light skinned, but with the darkest of Jews being dark from working out in the sun a lot unlike the wealthy class Jews regardless of what the normal color is.
Finally there's Revelation 1:15 in which a vision of Jesus to John has Jesus with bronze feet -- suggesting that Jesus ain't lily white like Jews in the Charlton Heston movies. But the verses surround it say that Jesus' eyes were fire and a sword coming out of His mouth. So clearly the bronze feet aren't meant to identify His skin color but be symbolic of something entirely different (Jesus is mightily standing steadily like something with a bronze foundation).
So much for getting an answer about Jesus' skin color from the Bible. The Ashkenazi Jews today are pretty white, but I'd guess that generations of living (and probably some intermarrying) with Europeans might have lightened the herd up a bit and, therefore, shouldn't be indicative of what Jews looked like thousands of years ago. For that matter, Acts 2 describes Jews as being at Jerusalem for the Pentecost celebration (at the end of the Feast of Weeks) and they spoke other languages (marveling at the disciples being able to speak the languages they recognized from the nations Jews had been living in for generations). No one would be surprised if by Jesus' day the Jews were a bit multi-colored by then. In fact, Joseph and Mary had been living in Galilee before Augustus' decree forced them to go to Bethlehem to be counted for taxes (even Jesus' family experienced the wrath of taxes). History outside the Bible suggests that Galilee had a lot of Itureans living there by then and intermarrying with the Jews for generations. What color were they? Where they an offshoot of Arabs? Or Arameans? Only God knows, and whatever their color only God knows if they contributed to Mary's gene pool.
If you are so concerned about his skin color—you do not know him or his message. Look within your heart and cast out all racists thoughts. His message is what counts all the rest isn’t important. Love ye, one another. Jesus didn’t say love only those that look like you.