Help! I simply cannot keep basil alive.

Ok, so currently I have two basil plants, purchased already-grown from Trader Joes about a week apart. I repotted them in slightly larger pots - I keep them on our balcony (protected from wind - about 6 hours of sunlight a day). I water them lightly 2-3 times a week. But they ALWAYS start to turn brown at the base of the stem, it creeps up and eventually kills the plant. I have a photo of one of the plants I'll link to below... This has happened in about a month.

I've tried growing them from seed once and had the same experience. Any ideas on what I'm doing to kill my basil? I'm a monster!!! arrrgggg

Thanks in advance for any help!!

The Basil Killer

Here is a link that might be useful: {{gwi:904017}}