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The picture is blurry and not close enough to see the insect clearly so, I am going to guess that, if the object looks like "white bits", it could be spider mites or whiteflies.

Unfortunately, your photo did not come through. I am unable to see the damage. Purple spotting can be phosphorous deficiency, pH swings, or disease, but it will depend on how it looks and where it occurs.

In the meantime, this article will help you with the care of Basil:

https://www.gardeningknowhow.com/edible/herbs/basil/how-to-grow-basil-plants.htm

Unfortunately, this happens as the plant ages. There is little that you can do, as it matures into a small shrub. What you can do is pick, only, new growth and keep flowers pruned away. Older leaves will be bitter, as will young leaves when flowers are present.

Here are some articles that will help:

https://www.gardeningknowhow.com/edible/herbs/basil/bitter-tasting-basil.htm

https://www.gardeningknowhow.com/edible/herbs/basil/blooms-on-basil.htm

This is the normal progression for the shrub. Continue topping it for more production, and to keep the flowers off.

They appreciate feeding, on occasion, as well.

These article will help:

https://www.gardeningknowhow.com/edible/herbs/basil/how-to-grow-basil-plants.htm

https://www.gardeningknowhow.com/edible/herbs/basil/information-on-how-to-grow-basil-indoors.htm

There is no way to cut it that leaves no woody stem, unless you cut it to the ground. There is no reason to do that, though.

Just prune it as you would a small shrub, but use what you prune as food, instead of throwing it away. Keep flowers trimmed if you want it to be less bitter. It is possible to keep it pruned into a small shrub, and continually use the new growth as seasoning. Continuous pruning will keep it alive for as long as you give it attention.

https://www.gardeningknowhow.com/edible/herbs/basil/bitter-tasting-basil.htm

Constant pruning will be necessary. I prune mine every couple of weeks. Mine is mostly ornamental, at this point, though. Once it has grown into a small shrub, it tends to stay a little bitter.

https://www.gardeningknowhow.com/edible/herbs/basil/information-on-how-to-grow-basil-indoors.htm