Thank you for the photo and for contacting us. The insect you have seen is probably the four lined plant bug. The damage fom this pest is being seen now. It is an insect with piercing, sucking mouthparts and that is what the brown spots are-places the insect has fed. It also looks like some of the spots are getting a clear center and will fall out. The bug moves very quickly so the fact that you saw it and described it as striped makes the diagnosis easier. These insects are only active for about a month so if the damage is not too great and you can tolerate it for that time your basil should recover and make great pesto! If you feel the need for control, rather than ignoring the pest, here is a link to a website with suggestions: www.vegedge.umn.edu/vegpest/CUCS/4line/htm.