Overall your basil looks great and you should be commended for growing this inside, which is a challenging task. The insects look like whiteflies or mealybugs :(. You can wash these away with lukewarm water.

Thanks Chervil! I will search these forums to see what natural pesticide I can create in my home.

Did you take a look at my other pics? My other basil babies are being attacked quite aggressively.

Use a pyrethrin.

Wow, I am envious of your basil! Mine was doing great outdoors this summer until it got too hot, so I brought it inside. Needless to say, only two branches are left standing and it had gotten pretty big this year. Now I am motivated again to try growing basil, this time exclusively indoors.

Thanks Dottie! My plants are growing very fast but they are still being attacked.