The majority of answers have suggested that breadth doesn't hurt you, but I have a more pessimistic view, based on my personal experience. I find that it is quite common for people to judge your research ability by what they consider to be your best piece of work. So if you're faced with the choice between

focusing on a single area, and publishing two papers in that area, one mediocre and one outstanding; versus

working in two different areas, and publishing one good paper in each area,

then my belief is that you will usually be perceived to be a better researcher in the former case. Now, it's true that the above dichotomy may be a false one. Perhaps your skills and temperament are such that if you choose to focus in just one area, you get bored and can't produce any good papers. But to the extent that you are faced with that choice, then I think choosing to focus on producing one really outstanding piece of work will generally be better for your reputation as a researcher.

Having said that, I don't necessarily advocate shunning breadth (again, assuming you actually have a choice). First of all, as others have pointed out, depending on what job you're after, your perceived status as a researcher may not be the most important factor. Secondly, in the long run you are almost certainly better off working on things you're interested in, rather than pursuing things that you think will impress other people. Of course, some compromise may be necessary since you do need to find employment somehow. But if your natural temperament inclines towards breadth, then ultimately I think you'll be more productive if you pursue your natural inclinations instead of forcing yourself to specialize in a way that you find distasteful. One advantage of breadth is that you have a good chance of finding connections between areas that specialists would never have seen. It may take you a longer time to produce that outstanding paper, but it could very well end up being even more outstanding than any paper you could have published quickly by specializing.