Is there any fur or mould on the undersides?

I don't think it's disease. The flecks on the top sides looks like insect damage.

This plant looks scorched. when you underwater basil too much it goes a little limp, combine that with extreme heat and extended sun exposure, the plant is really at risk.

is the plant in full sun all day? maybe move it to another window with only partial sunlight or leave it in the current window and remove it to a table or counter mid day.

keep it watered, maybe every other day as long as there is good drainage you shouldn't have problems.