There are several possibilities for insects chewing the basil, and while we can't rule-out slugs/snails, as they can feed on a variety of plants as well, we do not see any tell-tale slime trails to suspect them. Midsummer leaf-chewing insect candidates include caterpillars, crickets, young grasshoppers, young katydids, earwigs, and lots of beetle species, some of which only feed at night. You could inspect the leaf undersides, both during the day and at night, to see if you can catch a culprit present. For example, hiding among the leaves of two basil and one sage plant in a balcony planter of mine that was being chewed were 3 green caterpillars that blended-in very well. (I was not able to ID them to species level, though they were some type of moth.) Two fell off as the basil were being planted (late in the season), and one was stretched along the underside of a sage leaf next to the basil.

Insecticidal soap only works on direct contact with the pest; that is, any insect that encounters spray residues after they have dried will not be affected. Therefore, it's best to use it only when the pest is observed and can be sprayed directly, which is probably why applications thus far have had no effect. If you can't find the culprit under leaves or at night to remove by hand, you might be able to use a product using the active ingredient spinosad, assuming it is also labeled to use on edible plants. Follow all label directions for use in that case. As an alternative, the active ingredient Bt can work for caterpillars (certain strains of Bt, which will be labeled as such), but only when the caterpillars are young, as it becomes much less effective as they mature.

For future seasons, covering-up any herb or vegetable that doesn't require pollination just after planting can help to exclude pests. Insect mesh netting and row cover are two options, with the former being more airy in the summer heat and the latter being more insulating (by a few degrees) in the spring and fall around the time of frost, extending the growing season by a couple more weeks or so. Granted, it might be tedious to uncover and re-cover plants each time you want to harvest, but it's one option to minimize losses to chewing that won't rely on repeat pesticide applications, even if organic. That said, perennial plants should rebound well without intervention (sage, for example) while annuals like basil will still be productive, but might have a reduced harvest if leaf-chewing insects are abundant at some point during the summer.

Miri