Contrary to popular opinion, fitter people can actually sweat more than unfit individuals. This is because the more well trained you are the more physical work you can do, and therefore the more heat your body can produce as a by-product of an increased metabolic rate. To cope with the ability to heat up faster, your body adapts its sweating response to kick in earlier, and causes your sweat glands to secrete more sweat to allow for more effective cooling overall.
The misconception that unfit people sweat more comes largely from the fact that if you get two people of differing fitness levels to do exactly the same physical task at the same intensity (e.g. walking up a large flight of stairs), those who are unfit will be working much closer to their maximum capacity to do it, so they will get hotter and sweat more. Whereas the level of exertion will not illicit the same kind of response from someone well trained, and who finds the task physically easy.
Genetics also play a big role in how much you sweat – some people do just have higher sweat rates than others, and it is, unfortunately, not something you have much control over.