Depends on the game, Final Fantasy for instance, I almost always side with the originals because I feel the early games had pretty solid mechanics and often the changes to the game cheapen the original experience, on the other hand, the Dragon Quest remakes make the franchise finally playable and I chuckle to myself how often people praise the series having only played the remakes not realizing how much of the UI has been updated to make it less of a pain in the ass to get through. I had a similar experience with Zelda: OoT3D which I felt tighten the controls enough to make the game feel less clunky than the original.

With all that said, I don't mind remakes, my issue is that I've reached the point where I want the developer to seriously remake the game, not just update the graphics and maybe fix a few problems, I want developers to bring something new to it, which is why I do really like the DS remake of FFIV as it brought a lot of new content to the original and made it feel like a fresh new experience, I wish they had actually gone a bit further but who knows. Even if the remake sucks, its not like you don't have the original to go back to.