I never put any thought into it. I guess I'd call it a western first, sci-fi second. Keep in mind that I haven't watched it in a long time.
The Adventures of Brisco County, Jr., on the other hand, I have seen recently and consider to be sci-fi first and western second.
Anybody besides me remember the TV series Legend? It starred Richard Dean Anderson and John de Lancie. It was very much sci-fi first and western second. It was also wonderful.
I never thought about it much, but I guess it's science fiction to the same extent that any James Bond movie is science fiction.
On the other hand, in my personal library, I have The Best of Science Fiction TV*, and author John Javna and his panel of experts ranks The Wild Wild West as No. 8 in the top 15.
* Companion books in the set: The Best of TV Sitcoms, and The Best of Crime and Detective TV, co-written with Max Allan Collins.
I have two of those books. The amazing thing is that Lost In Space, a focal point in the Sci-Fi episode was ranked #6 in the Worst Sci-Fi show list! But then it was published in 1987(!!) so opinions may have changed!
Of special note here, Doctor Who, with only seven Doctors at this point, was #5 Best Sci-Fi TV!
Wonder what the rankings would be now?
I didn't discover the series until high school, but do love it and have all but the last season on DVD now.
The Wild, Wild West was as much a product and reflection of its time as Star Trek, The Man from U.N.C.L.E.; etc.
It is a fond remembrance of the past and I would classify it as a western because none of the gadgets on the show were as far fetched as you might think. Granted, no one back when the series was supposed to have taken place actually thought of such devices, but everything was technically possible, given the science of the day.