Door leaks

My new 2004 Cooper lets quite a bit of water into the door frames when I wash it. In and of itself, this doesn't bother me much - I can easily wipe it away. But what does bother me is that water drips down from the front door seams and is starting to stain the black plastic trim on both sides. My Aerospace 303 can't clean it off.

Does this sound like a fault or is it something Coopers just do?

Thanks in advance for your advice!