This article from Microsoft Answers might help. A lot will depend on whether you can start the phone and whether the problem is just that the display is broken or whether there's more to it and the internals aren't working either. The article does, though, say "With Windows Phone, all your contacts are stored in the cloud.", which is what you want to hear. Even if you can't access them on the phone, it sounds as though they aren't lost altogether. The same paragraph provides a link to how to access your contacts in the cloud.
Editewd to add: I've assumed that this is a Windows Phone device. If it's an older Symbian device, you need Nokia PC Suite and there are various downloads for it. They're all independent sites, so be careful.
