Don't know where you are located, but basil likes it hot similar to peppers.They also like sun.Have you considered some kind of light for them if you can't place them in sun.As for your question : 1.clay pots will dry out really fast if you watering once a week may not be enough.2.perhaps you bought them already with a problem? 3.did you clean your pots before planting them? 4. Did you use good soil?
All above are important.Having said this I would just part with them before you spread whatever they have to other plants.Wait a bit and buy again from a reputable nursery and plant directly outdoors protected area with lots of sun.
This is what I would do.

Thanks Scarleta. Today some leaves have darkened areas on them. Not so much black spots, but kind of browned areas. Anyways I put the pot outdoors, and will bring it in each night. If nothing works I will buy a new plant. BTW I will in Bothell WA.
Vaishali