I was more familiar with them as The Twin Towers, but I knew of them, sure, and I’m slightly younger than the OP, and live nowhere near New York.
I worked on the Lehman Brothers trading floor for 2 weeks during a business/training/orientation trip when I was based in Hong Kong. So, yep, knew exactly what they were.
Yeah, if it weren’t for that aforementioned school trip I’m sure I probably wouldn’t have heard of them. I was 5 during the 1993 bombing and have absolutely no memory of it; actually, I still don’t really know what happened. The school trip and 9/11 took place when I was 13. I think that’s about the youngest I could have been and still had a pretty good comprehension and vivid memory of what had happened, because most people younger than me don’t seem to.
No, I did not know what the World Trade Center was before September 11, 2001:
Saw them first in '81 when I was a kid, and took a picture. During the attacks–for the first one a co-worker told me someone had blown up the Empire State Building.
On 9/11 I was in class and a woman showed me all the news on her pocket computer. I didn’t believe the towers had fallen because I had seen them and to me they’d looked sturdy. I couldn’t envision them coming down.
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I had seen them a million times just from imagery in pictures, television and movies that were shot in NYC. And I had heard mention of the WTC many times as well, I just never put 2+2 together until the bombing in '93 (I just turned 20 at the time).
So by the time the 9/11 attacks occurred, I knew fairly well what they were.
My brother lived about 6 or 8 blocks away from them. I was aware of them.
I’ve known of them since they were first built: I remember when they opened, they were the world’s tallest buildings at that time. And since then I visited them twice, and went to the observatory each time.
And I’ve never lived anywhere in New York state.
Sure I did. I’ve lived my whole life in California, but when I was 12 my mom took me to NYC and we went up in them. My husband and I visited in '99 and decided not to spend the money to go up, which I rather regret now.
Also Snake Plissken landed on the top of one of the towers in Escape from New York.
I did live in NY from 1978 on, but I had heard of the towers before then. I could understand people who were born after it was built, and did not live in the region, not knowing about it before 9/11.
For what it’s worth, I knew people who did not know the phrase “World Trade Center” but were definitely aware of the “Twin Towers”. They were 12, though.
As a side note, I think the Chrysler Building is slightly more recognizable than the Empire State.
I can remember back when I was a teenager and one of the towers had a big fire. It was widely reported because this was around the same time the movie The Towering Inferno was in theatres and that was about a skyscraper fire.
I don’t remember when I first heard of WTC but am sure I was already an adult at the time. 
I’d find it slightly odd not to know of the Twin Towers … but others would find my own ignorance of contemporary American culture to be very odd(*).
Trivia tidbit: Bangkok’s “World Trade Center”, “third largest shopping complex in the world,” was the scene of greatest destruction during the terrorist siege of Bangkok last year. (Though from Wiki I now see its official name changed from WTC before its gutting.) I don’t think the terrorists targeted it due to the WTC name connection, but simply due to the political alignment of its owner.
(* - As one example, the term “hip-hop music” was still unfamiliar to me in the mid-1990’s. And that was before my becoming a permanent ex-pat.)
There is also a (smaller) World Trade Center in Baltimore. There was a reception there for a conference I was at a month after 9/11. It was very creepy.
Being a New Yorker, and then New Jerseyan, I of course knew. I’ve been inside, and under, and we stayed at the Vista Hotel next to it once.
Another “of course I knew.” I’d been to New York and seen it as a kid, but I knew what it was even before then. I can’t remember ever not knowing what it was. They were the tallest buildings in the world for a while.
Another “of course I knew,” but my father in law works at a call center in Arkansas and he says that on 9/11, none of his employees had heard of it.
British, I knew what they were. For a while one of them was the tallest building in the world. And King Kong of course. I was so impressed that I made a trip to the observation platform on a visit to NYC - on 9/11/97. Something that freaked me out despite being a coincidence.
ETA: sorry missed ‘Americans’ in the thread title.
I knew the twin towers as the World Trade Center and could pick them out of a photo of the NY skyline since at least the 1993 attack. I remember learning to connect the name to those particular buildings, but can’t recall when that actually was.
34 year old female, born and raised in Australia (I’ve never been to the US).
No.
But I was only 11 years old and living in Arizona when the towers fell so I don’t think I can be blamed.