I have a similar problem with these tiny white flies. It considerably weakened some Shrubby Cinquefoil (Potentilla) seedlings I was growing in pots in the garden and is still feeding on them. Although you haven't posted a photo, given your description, I'm still confident that the insect that attacked your basil is the same as the one I have been doing battle with, whitefly (almost certainly the Greenhouse Whitefly variety, Trialeurodes vaporariorum), particularly as basil, together with tomatoes and cucurbits, is one of its favorite foods.
Whiteflies are difficult to control, both chemically and biologically. I have tried spraying with a soap solution, which is fairly effective, but only for a day or so, after which they return and the plants have to be sprayed again. There is an excellent article about them here, and a very useful organic pesticide chart here.
I've done some research in an attempt to find out whether there is a more effective method of controlling this pest;
The flies [1-2mm in length] can be found on the leaves of plants - both on the upper and lower surfaces, but mainly on the latter.
These are the adults, which feed on growing shoots and lay eggs that hatch into tiny white scales [larvae] that remain attached to the underside of leaves. Like aphids (greenfly, blackfly), whitefly are sap-sucking insect pests.
This sap sucking weakens the plant and may introduce plant viruses that further weaken the plant, and may even kill it.
Unfortunately, whitefly don't go round in ones or two - they go round in hordes of hundreds and a severe attack can severely weaken a plant. You will often see a 'cloud' of whitefly flying away from a plant whose leaves have been disturbed.
Source: Garden Forum - Pest Watch
You say the flies “spread out like snow-flakes” when you spray your plants, and this is exactly what happens when I spray or disturb mine.
Here is what I suggest. You could take only one or two or, preferably, all of these measures, depending on how bad the infestation is and how much time and money you're prepared to spend:
Buy some Yellow Sticky Aphid Whitefly traps, or make your own, plunge your plants in a bucket of water for a minute or so, drain them and then place the traps around them.
Vacuum the plants with a small hand-held vacuum cleaner, preferably in the early morning when it is still cool and the flies are sluggish.
Place your plants in a mini-greenhouse, and then introduce a parasitic wasp called Encarsia formosa (it doesn’t sting) which, it seems, is highly effective in controlling whitefly in a confined space, and is used by commercial growers. Live Encarsia formosa eggs are available here. I have just bought a mini-greenhouse and this is a method I intend to try myself. Although my plants are outdoors, native predators have failed to control the whitefly and, as their numbers have increased, spraying with a soap solution has proved less and less effective.