I'd first recommend that you ask her what she is afraid might happen during storm, as the best way to help her is to help her understand that the things she fears happening are either unrealistic or unlikely. That will at least help her mentally know she doesn't need to be afraid.
Beyond that, you can try to point out the beauty of a storm. Show her videos of storms where she can watch the lightning and rain and see how beautiful it is (make sure you preview the videos and that they are not about catastrophes or anything, there are soothing videos with storms that might help with this).
Discuss how storms help people. Take her out to play in the puddles after a storm (even an 8 year old loves to jump in the puddles).
Perhaps get her a rain gauge and put it outside, have her try to guess how much she thinks it is going to rain, and then when the storm is over she can check and see how close she was.
When she watches the weather channel, watch with her and discuss what she is seeing. What is she hoping to find out from them? Can they tell her what she wants to know? How do they get their information? How often does their information change?
If the weather channel is adding to her fear, ask her to not watch it during the storm (this may make her feel more frightened though as she might feel she has even less control).
Discuss what types of storms cause real damage and which do not (most don't). When she is getting frightened, ask her what type of storm this is.
If tornados are her main fear, she might be frightened that a normal storm will turn into a tornado. Discuss how tornados really start and what we can do to keep safe during them and any other storms she may fear. Try to emphasize the control that is in her hands. My guess would be that storms are probably making her feel like she is in danger and there is nothing she can do about it, so the more control you can give her the better.
If she continues to be very frightened, it may be time to get some professional assistance from a psychologist. They can help her work through the fears, and prevent it from becoming a full blown phobia, which from what you describe she might be on her way towards (the fact that she calms down when you go in the basement makes me think she's probably not there yet, and it is easier to catch her before she works herself up to that point).
I hope that helps, if I can be of further assistance, please let me know. If this answer has helped, please click the accept button.