I think it highly depends on who does it and especially HOW. A perfect example I'd like to share is the experience I had when playing Pokemon Blue and completing it, and then playing its remake, Fire Red.
To me, Fire Red is the perfect example of a remake well done - not necessarily perfect, but definitely well done. A first reason for that is because the Pokemon Company knew what was in the old versions and brought everything back from it, except with updated graphics, soundtrack, sprites and options - such as little things like the Running Shoes, which make long-distance travel so much more enjoyable than just walking around at a pace so slow that it bores a player quite easily) when you can't use the bicycle.
Then, ASIDE just bringing everything back, they updated certain moves, added new ones along with a brand new Ability system recycled from Ruby and Sapphire, as well as a big subquest after defeating the Champion (who's remastered theme is by the way freakin' amazing, that's one of the reasons why I loved Fire Red with a passion after completing it), and then they even added MORE than 150 Pokemons for us trainers to capture. Heck, they even raised the level of the Elite 4 and Champion after all this just so we could challenge them again, AND added the VS Seeker, which allows a trainer to fight trainers over and over again, AKA gaining more money and EXP than if fighting against countless Wild Pokemons.
So there you go. To me, a remake can be a good thing as long as the original product is respected and shown in the remake, but also if new things are added in conjunction with what was before. A harmony between what used to be and what is available now. I could also give HeartGold and SoulSilver as a great example (it even had the option to play the entire gmae with the 8-bit soundtrack for the pure nostalgic ones).
In any case, my voice on the matter would say that yes, a remake CAN indeed make the original product better, but as I said in the beginning, it really depends on what's in it. Rayman 2 for example had a lot of remakes and ports, and not all of them are great.