A photo of the beetles feeding on your plants would help with a positive identification. 

But from your description and the fact they feed at night, they sound like Asiatic garden beetles. Adult beetles are mostly a nuisance, but they can chew the edges of leaves causing leaves to look ragged. At this point, treatment with an insecticide is not recommended. Go out in the evening with a cup of soapy water and knock the beetles into the cup. The basil is still usable. This pest should not be around too much longer. If this is not the culprit please send us a photo. 

Please see the following link for more information,

http://bugoftheweek.com/blog/2014/7/21/midnight-marauders-asiatic-garden-beetles-imaladera-castaneai?rq=Asiatic%20garden%20beetle

Deb