Here's my take.
Remakes/Reboots
In general, those films don't affect my enjoyment of the original material because they don't "change" the original material for me. It's like a different continuity, different universe. So, in regards to the original product, not a big deal to me.
What bothers me about remakes and reboots is that it is far often more about pure marketing than about actually good content. Sometimes it works out great, but much of the time it's just slapping a brand name on otherwise ho-hum material.
If I could convince Hollywood producers to think like this, here's what I'd say: If you're thinking about doing a remake, stop and ask yourself whether, if you removed the branded material, your movie would be worth a crap. If the answer is "not really, no," then don't make that movie.
Sequels/Prequels
These I have a problem with. I've tried to explain it before, but my bet is that if you don't get what I'm about to say...you just won't get it. It'll never make sense.
For me, a bad sequel or prequel CAN ruin an original, if only because, as OB described, it "fills in the blanks" or tells me "What happened next?" Moreover, I can't put the bad sequel/prequel out of my mind when I'm watching the original. I know what happens next now or what happened before, and so I'm thinking about that while watching the original. If those sequels or prequels are lame, then the disappointment I feel at those bleeds into the original for me.
Sometimes I can get enough distance from them to still enjoy the originals, but often I can't. This is the reason why I haven't popped the Matrix into my player in probably five years. Maybe more. I just have no desire to watch it, because while I'm watching it, I'll be thinking of how I think the next two movies just suck. With other movies, I can sometimes come back to them years later, if the originals are good enough and stand well enough on their own, but the experience will never be as enjoyable to me as it was before I saw the next entry.
I know this is a weird thing about me. I don't understand it, myself, even. And in many cases I WISH I didn't function this way. I'd love to be able to watch a bad prequel or sequel and be like >shrug< whatever. Doesn't bother me any, but if it's a movie I really really love, I know that's not gonna happen. I've learned this the hard way. It's happened with too many franchises for me. So, now I just...opt out. I don't bother watching sequels/prequels if I'm worried they're gonna suck, particularly about favorite film franchises. I wish I didn't have to, because sometimes I'm curious about a film, but I know that if it sucks, I'll just end up enjoying the originals less. Or at least that there's a chance for that to happen, so...I avoid 'em.