I don't know Swedish buggies, but if it was here in the US, I'd guess spidermites. I don't know why the basil would be unhappy enough to get spidermites... You COULD throw the plant away, because you won't be wanting to eat the leaves after you get rid of the mites... or, here's MY spidermite cure... 4 liters water 15 ml dishsoap, 15ml cooking oil and 15ml alchohol or mouthwash and maybe 15ml hydrogen peroxide. or you can buy a soapy pesticide. And you have to respray for like a week because the eggs will hatch.
:(

Try spraying with neem seed oil. Its quite safe and can kill most any insect pest that is feeding on the basil. The spray is safe enough for use on herbs. Simply rinse off the leaves when you need some. If its in partial sun, that may one cause for the problem. Mine grows quite well in a sunny location from sunrise to about 3PM. Never saw anything kind of leafe damages though. If you spray the leaves with something, that may be the cause.