Protect with row cover.

Sorry, but row covers are definitely the way to go. Unfortunately, they won't save your current crop.

Purty Simple-find the little buggers in the leaves and smash them--look for eggs--little white dashes on the underside of the leaves and rub them off. Murder can be very satisfying!

I had them in my spinach. I hand squished them before they could form very big tunnels. Then the ants showed up and took care of the problem. They attacked the larvae and ate the eggs - no more leaf miners. It was wonderful! I'm not sure, but I wonder if the squished leaf miners attracted the ants and started the whole process. Now, any ant colonies are welcome in my garden!

Bellatrix

Also, watch for the Leaf Miner Flies as they emerge from the soil in Spring, and deal with them organically--they're easy to kill (they're about half the size of a house fly). Once they've laid eggs in the leaves of Chard, Spinach, Turnip leaves, Beet leaves, etc., (you'll see the tunneling trail of the "maggot" (larvae) throughout the leaves), pick the leaf and hold it up to the light, until you find the larvae. Squash it with your fingernails (they're harder to kill than they appear--tough little buggers), and discard the leaf. Doing this, of course, won't save those particular leaves, but it will wipe out the next generation of Leaf Miner Flies, if you are dilligent. Don't continue to grow Chard, Spinach, etc., in the same spot year after year--it's an invitation to infestation--rotate that type crop. I'm aware that flies can fly, and possibly STILL find your crop, but it's just too easy for them to hatch out of the soil, and be right AT the crop they'll infest.