I did this once years ago and followed an example from Wilki How site

Fill 3/4 of your container with soil. Allow the soil to remain loose; do not pack it into the container tightly. Tightly packed soil will make it difficult for your basil to take root

Moisten the soil. Do not over-saturate it, but add enough water to make the soil mildly damp.

Sow a few seeds an inch or two (2.5 to 5 centimeters) apart if using a round container.
Scatter seeds along the length of the container if using a long window box. If using this method, you must thin your basil out once seedlings sprout and reach an inch or two (2.5 to 5 centimeters) in height.

Moisten the soil with a spray bottle. Germination should occur within a week.

It worked – I don’t grow it anymore dose not get used – But good luck

I read that major cause of black, dying basil leaves is cold weather. Temperatures below about 10-12 deg. C will start to damage basil leaves,

I would have thought that at this time over year basil needed to be on the kitchen window sill rather than outdoors.

It is on my kitchen window :)

Is it too damp and/or gets steam from cooking?

MG's probably hit the nail on the head, possibly too damp a situation, not enough ventilation and it's a fungal infection. Try it somewhere lighter, more airy, and remove any infected leaves.