Jesus was a Jewish man from the eastern Mediterranean 2,000 years ago, so the logical conclusion is that he looked like the way the typical Jewish man from the eastern Mediterranean 2,000 years ago looked like.
The closest example would be Sephardic Jews without a history of intermarriage, although similar sects such as Druze, Alawaites, Maronites, etc. would also be helpful. Whether that is "brown" or not is in the eye of the beholder. He certainly was neither European (as was commonly shown in European paintings) nor Arab (which is what I think some people think of when they say things like "Jesus was brown.")
Anyways, most Christians historically have simply drawn Mary and Jesus to look like people from their culture. So, Italians made pictures of Mary and Jesus that looked Italian, Spaniards made pictures that looked Spanish, Ethiopians made pictures that looked Ethiopian, Japanese made pictures that looked Japanese, etc. (I think I saw one in a museum by a Danish artist that gave Mary red hair actually.) This is not recent, the Virgin of Gaudeloupe (the "brown lady," brown being mestiza here, not Arab (or Shephardi Jew) and not European Spanish either) was done in 1531. There are Japanese Virgin Mary's from hundreds of years ago, and of course the Ethiopian church is one of the oldest there is. There's nothing nefarious about this, except insofar as the European version crowded out other local versions in certain places (though understandably there can also be confusion or tension where a society has become more diverse and thus the pictures look dated).
This is a great example where the current trend toward iconoclasm ("let's hate on or even destroy cultural artifacts to show how woke we are") is totally wrong. We should simply encourage people to draw Jesus and Mary however the heck they want.
Middle-eastern, hispanic, white, native american, sub-saharan african, asian, indian, aborigonal. Whatever. And of course some depictions that intend to be as historically accurate as possible would be nice too, but that's not necessary for religious iconograpgy.
By the way, since this thread is about trying to modify language to be less offensive, many people would be offended by someone who is not religious (no idea what your background is, just going on numbers based on this website) and who doesn't seem to know much about the history of religious iconography telling people they need to pull down their religious statues. I'm not that easily offended, so whatever, but I know many people who would be. Remember, respect is a two way street.