Keeping Fresh Basil and Ginger

I received an Aerogrow garden and am very excited. I absolutely love basil. Also ginger. I do not know how to preserve these herbs either in the freezer or refrigerator. I have tried various methods but the fresh basil turns black and yucky and the ginger turns rancid.

You did not mention the quantities you are growing and whether you will harvest all the basil at once. Fresh herbs are more fragrant and flavourful than preserved so I will suggest to use fresh basil as often as you can. Basil does not dry well and has a limited storage time in the fridge.
You could do the following for storage in the freezer. Fill an ice cube tray with chopped basil and fill with water. the cubes can then be kept in freezer bags. Do not use too much water, though. The same method can be used using oil instead of water. See how this will work for you.
The ginger roots keep quite well in the fridge and I would not freeze them. You mentioned that the ginger turns rancid, but keeping the ginger root in a small jar, will prevent it from absorbing any other smell... Candied ginger is a delightful way to preserve ginger and will keep well for a length of time. Let me know if this method would interest you.