I am reading that no to pesticides, but try some cayenne pepper mixed in soapy water... good luck! I don't know the recipe, but you don't need much more than a drop or so of soap... probably could use a lot of cayenne... and remember, WASH THE BASIL before you eat it! :o)

We just discussed the different leaf-miner signs in class yesterday (Plant Problem Diagnosis): "Blotch" leaf-miners are just as you discribed and then there are "Serpentine" type leaf-miners that have sqiggly snake-like burrows thru the leaves.
Look at the following images of both types of damage.

Parasitic wasps do attack the miners, but I don't know which or what to look for that says a burrowing miner that is paratized looks like?..I can ask my instructor today.

Vera

Here is a link that might be useful: Leaf-miner

Here are two recipies for cayenne spray:

Pepper Spray
½ cup finely chopped or ground hot peppers (or 1 tbs cayenne)
1 pint water
a few drops castille soap
Combine peppers and water. Steep for 24 hours. Use as is for a soil drench application, or strain and add soap to use as a foliage spray.

All-Purpose Insecticide

1 bulb garlic
1 small onion
1 tablespoon cayenne (red) pepper
2 tablespoons liquid soap
1 quart water
Spray bottle

Mince garlic and onion; mix with tap water; add cayenne; let sit for one hour, then add soap. Mix well. When foam subsides, strain into spray bottle. Keep refrigerated. Good for 1-2 weeks.

We too were discussing leafminers in basil just recently. Like one of the members said, "By the time you see them it is really too late to do anything." I just keep removing the damaged leaves, and I have found eventually that they just finally go away. I have never found the sprays to help.