Really, I think it depends on the game, and the extent to which they're overhauling it.
As a general rule, I have nothing against remakes. If the game is being remade fully, the chances are it's been out for at least two generations now, and it's nice to see these things brought up to speed; remakes usually mean glitches are ironed out, gameplay is more streamlined and, best of all, additional content. Whilst remakes are all about making money, a lot of the time they justify the re-release by turning it almost into a completely different game: using Pokemon HG/SS as an example, they added so many additional features to that it was unbelievable, remixed the music, updated the graphics, and even packaged it with a Pokewalker reminiscent of the Pocket Pikachu. Whether you like it or hate it, that is a remake done right: it keeps the formula of the original, but establishes its own identity by adapting it to fit the parameters of the current generation. It differentiates itself enough to be worth buying for more than just the nostalgia it generates in people who played it the first time around.
In the case of HD remasters, which are not technically remakes but are something I want to touch on anyway, I think their potential needs to be utilized properly: we need to see more games like Zone of the Enders being given this treatment, and less like the Prince of Persia trilogy. That is to say, older games that are extremely difficult to find for a reasonable price and haven't had quite as much exposure as other games may have. Games like Final Fantasy X, God of War, Splinter Cell and Metal Gear Solid, in my opinion, don't warrant HD remasters, because you can find them just about anywhere for a cheap price and really all they're doing is tacking on some trophies; it's a cheap ploy to make more money by developers who really don't need that money in the first place.
The 3DS remaster of Ocarina of Time, similarly, was just a shameful attempt to boost the sales of a handheld that had absolutely nothing in the way of worthwhile titles on it at the time. If games are going to be remastered, then the reason should be for more than just milking a franchise for all it's worth. It's a way to bring lesser known games to the front, not for popular games regarded as "classics" to make even more money than what they already do...or, at least, it should be. Alas, gone are the days when developers cared about anything other than making money. But I'd still like to see more HD remasters of games that are harder to find, and remakes of older, less popular games.