Easiest and safest way to kill it is to spray it with hydrogen peroxide. Just hit Wal-Mart or your local pharmacy and pick up a bottle. If you're lucky, you'll find some already in a handy spray bottle.

Pharmacy antiseptic grade H2O2 is at a 3% concentration, and is fine to spray directly onto the plant and the soil itself. Won't harm your peppers, and will kill the fungus immediately.

You can also mix some peroxide into the water you normally give the plants to help pep them up. About 2 teaspoons of 3% per gallon.

If you're trying to do organic composting and all that though, the dilute H2O2 watering mixture will kill off the good bacteria along with the bad, but if you're using normal miracle grow type fertilizers, you'll be fine.

Thank you for your helpful response Edy.

They are being grown organically so I guess I'll see what effect direct sun has first. If the fungus worsen though, I have no qualms about killing it with the peroxide. Pretty sure I have some already.

Spritzing the top won't hurt the bacteria cultures you have going below the surface. Its just mixing the peroxide in with your normal watering schedule that will do that.

Worst case you'll get is that you might steralize the top milimeter or two if you really soak it down with your spraying, but a simple spritz shouldn't affect anything but the surface fungus.

I just sprayed my peppers and broccoli I have in containers this afternoon. Out of a dozen pots, 11 had absolutely nothing happen. One broccoli though started fizzing around the stem immediately. Jackpot, I just nailed some dampen off fungus before it had a chance to kill my seedling.