What you are describing sounds to me like dodder. It is a parasitic plant that has not roots, and does not have chlorophyl. It lives entirely off of the host plants and eventually chokes them out as you describe. It can be a real problem with some herbs, and likely came in with the basil seed. You are best to throw out the soil in the pots as it likely now had dodder seed in it and it may continue to be a problem in the future.

Angus