Benjen was stabbed by a White Walker with an ice sword and left "to turn" but was saved with dragongalss by the Children of the Forest.
Benjen: I led a ranging party deep into the North to find White Walkers. They found us. A White Walker stabbed me in the gut with a sword of ice. Left me there to die. To turn. The Children found me. Stopped the Walker’s magic from taking hold.
Benjen: The same way they made the Walkers in the first place. You saw it yourself.
Bran: Dragonglass. A shard of dragonglass plunged into your heart.
Game of Thrones, Season 6 Episode 6, "Blood of my Blood"
I always assumed that he would have turned into a White Walker, he still seems to have conscious thought and in my opinion looks more like a White Walker than a wight. However, it's unclear how much of this is down to what the White Walkers did and what the Children of the Forest did.
Benjen does state he is not fully living anymore and implies that he is "dead" which would put him more into the category of would have been a wight.
Benjen: The Wall is not just ice and stone. Ancient spells were carved into its foundations. Strong magic to protect men from what lies beyond. And while it stands, the dead cannot pass. I cannot pass.
Game of Thrones, Season 6 Episode 10, "The Winds of Winter"
However, later on Benioff states that he wasn't really dead or alive but somewhere in between which could put him into either camp. We don't know enough to state if it was the White Walker's or the Children of the Forest's magic which put him more into the "dead" side.
Benioff: For Coldhands, it’s almost kind of a relief in a way because he’s been trapped in this kind of purgatory state between life and death for quite some time and, like so many in the show, waiting to find out what his purpose is. Why is he still alive when he should be dead?
Game of Thrones, Season 7 Episode 6 - Inside the Episode, "Beyond the Wall"
Is there any official statement as to whether Benjen would have been a White Walker or a wight had the Children of the Forest not intervened? I am not looking for speculation from implicit comments as they seem to be slightly contradictory and unclear due to multiple events having occurred but rather something explicit. That could be from a comment in the show, a comment from one of the supplementary materials or from an interview with someone in the show.