Did you have that basil problem as well? Whenever I had a basil after a few weeks it died either of too less or too much water. It was really hard to not dry out the plant while avoiding molded earth.
Two weeks ago I couldn't resist the green leaves at the supermarket and bought a plant again.
Some days later I visited my mom. She had a new basil in her kitchen and so I asked her if she knew that "basil-difficulties", too. Sure she did.
But a little while ago she had heard from a friend from a friend who was supposed to have the solution to it. Really!
I couldn't believe it and now I try to follow the two easy rules:
1) No water from above. Spill some water into the outer pot and let the plant soak the water for some while. Remove it after about 15 minutes if there's water left.
2) Never cut single leaves. Always cut a whole branch.
Give it a try and tell me about the results. I'll let you know, too.