Well considering the Director of Rocky and Karate Kid is one and the same (John G. Avildsen), I'd say the similarities are pretty clear, while it wouldn't "technically" be a remake.
It looks like the guy hit it big with Rocky, but the next 10 years has only a few projects, one of which being about some teacher who dates a student in a strip club called "Heaven"... so I guess he figured a return to form(ula) would fix his career... and it did, wahoo.
2 Karate Kid sequels later he did Rocky V in 90.... hmmm...
But sequels are a different story altogether- if a character has more life in him/her than in the movie, sure, throw a sequel up there.
Remakes, it depends. I don't mind remakes when they're GOOD. I mean, technically if anyone makes a book into a film, they are REMAKING the story. Hell, if you turn a SCREENPLAY into a movie, you're REMAKING it.
If you gave the exact same script to 10 different directors, chances are you'll get 10 completely different projects... don't' believe me? Check out last years IndieTalk Script to Screen Challenge... Nick and I made a script that was made by 3 different groups, and it's amazing to see where everybody differed on what points, etc.... the only thing separating those 3 movies from becoming labeled "remakes" is that they were made at the same time.
So really, a remake should be either called "an update" or a "reinterpretation". If it's an update, obviously someone thought "hey, i love this story, but it doesn't really apply to modern society" and then they go and 'remake' the story to tell it with modern devices to a modern audience. If it's a reinterpretation, it's someone out there going "holy moly that story really inspires me, but man, if I had done it, I would have done this, that, this, that and thus" and then doing it.
So I don't think it's necessarily a bad thing, but if the original turns out "worse" than the original, it's really up to the perspective of the eye of the beholder. I never saw the 1960's movie of the "Andromeda Strain", but I have seen the recent TV movie and I can tell you, it's alright until it falls apart in the last 10 minutes of the film... but if I had seen the original, I probably would have hated it at the 20 minute mark.
Either way, I think the motive of "why" something is remade is important. If it's for money, then PAH, I could care less... but Will Smith remaking the karate kid with his son seems like a more noble reason than money, so whatever...