I'm no bugologist, but it's definitely a bug of some sort! (And a lot of bugs you get are different from the bugs I get in my part of the world.) A spray with Garlic Spray should fix it - it's good for multitudes of bugs. For a more accurate ID of the bug, take a few affected leaves to a nursery and get the experts there to point you in the right direction.

My guess is that the plant is indoors. Indoor plants are especially prone to bugs because there are few predator bugs to get rid of them. A spell outside might help, but it does look like a plague of them to me.

I'm not so sure that this is insect (or mite) related. It appears very much like a fungus disease at the stage where minute fruiting bodies have occurred. The only way that I could be certain would be to see these leaves in person. A hand lens wouldn't even be necessary.

Regardless, fungal disorders as well as pests can languish in a state of semi-snooze when outside, but burst into high gear when exposed to a 'hostile environment', i.e. indoors. Basil is not a good candidate for window ledge culture, especially when acclimated to the outdoors.

More likely to be damage from water shortage -- the plant lost lots of those when you dug, then put it in a pot.

It looks a lot like red spider mite damage. Possibly with some mold fungus which has come in after the initial harm.

Here is a link that might be useful: