The potted basil plant that I bought from the supermarket keeps dying!

Bacterial Leaf Spot or Basil Shoot Blight This basil disease is caused by a bacteria called Pseudomonas cichorii. Symptoms of bacterial leaf spot are black or brown spots that appear on the leaves and streaking on the stems of the plant. Bacterial leaf spot occurs when infected soil is splashed onto the leaves of the basil plant. While there is no fix for bacterial leaf spot, you can minimize the damage by making sure that your basil plants have plenty of air circulation and that they are watered in a way so that the bacteria is not splashed onto the leaves.

Toss it and get one from a Nursery ( not a 'Big Box' store ); and water from the bottom; don't let the lower leaves touch the soil

https://www.richardjacksonsgarden.co.uk/grow-supermarket-basil/

Put two or three expired (or fresh) multi-vitamins in your plants 4 times a year. Before I retired from teaching school, I did it on report card day. My plants are huge, including a 20 year old Peace Lily.