Most likely a form of red spider mite, although these prefer hot, dry conditions, and you say you're growing hydroponically; this often means moist air, although not necessarily. Assuming it is spider mite, neem oil in a spray should do the trick, but it would be best to spray only one plant, or a few leaves, to see how the plant responds to the treatment. Neem oil spray can sometimes cause a problem for plants, especially if temperatures are high. It can also be used as a soil drench, but you're growing in water, and I'm not at all sure how the plants would respond if you added it to their nutrient solution.
UPDATED ANSWER: What spider mites do is pierce individual cells in the leaf surface to feed on the contents - this causes a brown speckled effect, and if penetrated often enough, the leaf yellows, curls and falls off. Given you're growing Basil, where you want to use the leaves, it's not particularly desirable to leave the mites alone. You can certainly try keeping the air damp and humid because they hate that, and a few spider mites won't kill an entire plant - the trouble is, they overwinter off the plants anywhere they can find, such as cracks in brickwork, etc, and come back with renewed vigour the following year; they are a real pest in greenhouses.