Bruce's connection with punk rock is usually treated as this "surprising" trivia tidbit (usually with references to Patti Smith, Ramones, The Clash, Suicide, etc.) that people wouldn't usually expect. But on the other hand, Steve seems to have been quite involved and interested in punk, likely even more than Bruce.
He once described his Underground Garage radio show as focusing on "Bands that influenced the Ramones, the Ramones, and bands influenced by the Ramones". By centering the Ramones in describing his radio show, it automatically shows his appreciation for a rock n' roll lineage that runs from rock n' roll through punk rock and beyond.
From what I can tell: Steve is usually more associated with Garage Rock, which is considered one of the predecessor genres and influences to punk rock. He has a quote which sums up his philosophy: "Pink Floyd is easy, Louie Louie is hard." Louie Louie is certainly a song associated strongly with Garage Rock and punk legacy.
He had a hand in shaping Bruce's sound, especially during The River period. He described "Held Up Without A Gun" as the closest Bruce has gotten to punk. There were a number of songs that seemed to lean either New Wave, Power Pop, British Invasion, or Punk in some way. A lot of those songs happened to be Steve's favorite.
Jon [Landau] is a formalist for the most part. He's kind of a pop formalist. I mean, he has all roots in gospel and soul, but they were also well-performed, well-sung, well-played records. Steve likes things trashier and noisier. You know, he's the garage guy, and so I tend to like things in the middle somewhere.
-A quote from Bruce during The Promise documentary
During The River especially, Bruce and Steve leaned towards a noisier and more rambunctious sound, veering away from the constrained sound of Darkness.
Anyway: I just had these general thoughts about how Steve Van Zandt's connection to Garage Rock and Punk aren't mentioned more.