How can you eliminate the minuscule white moths that live on but don't eat all of my herbs?

Those miniscule moths are most probably aphids, the bane of all gardeners. You can introduce a natural aphid predator such as ladybugs and attract them by planting fennel or dill. You can spray your plants with neem oil for an organic solution. You can also make a spray yourself by mixing in one quart of water 1TBS vegetable oil and two or three drops of any non-bacterial liquid soap. Hose plants first to remove aphids then apply spray. Hope this helps!

Thanks for the help, but I'm sure they aren't aphids. They are moths the size of poppy seeds.