I imagine a better root structure wouldn't help because it appears the water is gone. Am I wrong? No you're right on! Often the soils recommended here cause the exact same problems one wants to fix, droopy plants. In this case due to soils that drain too well. It's a balancing act. I myself think it ill advised to recommend fast draining soils when the problem is a watering problem, not a soil problem. In general you want a soil that retains water. Why so many new potting soils come with water retention products. Dry soils is a much bigger problem than overwatering. It's a huge leap to assume overwatering is the problem, if it's under watering and the soils are changed to a mix that dries even faster, the plants will most certainly die. First let's find out, is the soil lightweight, is it dry to the touch? Plants will droop from hot sun wet or dry and recover later. That is somewhat normal. The drooping is caused because the plant cannot uptake as much water as is evaporating from heat. It's a physics problem, and has nothing to do with over or under watering. So we have three possibilities as to what is causing the problem. Over watering, under watering or transpiration rate. If the latter as long as your plant recovers by morning it should be OK. Shading the plant could help, but sometimes the plant needs the sun, and so this is an acceptable occurrence as long as the plant recovers. I see this often with my blueberry plants. They like to be very moist, roots are near the surface and get plenty of oxygen so can take being moister than most plants. One needs to consider plant type. I'm not that familiar with basil, but the mint should not be drooping. It can take a lot of sun and not droop. It's hard to damage roots since they are rhizomes that interconnect. I suspect it needs more water. Based on that I would say that rashomon nailed the problem as under watering. Observe the plants, and try to determine of the three possibilities, which is the problem. We are all guessing based on our knowledge base, but you have the plants right there, be more observant of soil moisture and figure it out. So if you change the soil to an even faster draining one the problem very well could get worse.

Does the container that it's in drain quickly or might your plant be coming a bit soggy? OR...are you giving it tiny sips when you water it, so that the soil is actually very dry?

That is transplant shock as well as sunburn. Quite normal, and can be reduced by placing the plants outdoors in partial sun and shade a few days before they go in the garden soil. Basil is quite hardy, so it will grow back. One of my seedlings of basil had broken off an inch above the soil. I shoved the 7 inch long stem back into soil and buried the break. It wilted a day or two and then perked right up again. Two weeks later it went in the garden with the rest. Its doing just fine, except for a few older leaves that turned tan color. Quite normal, and will eventaully grow lots more new leaves.

I'd just pick off the affected leaves and dispose of them. Since basil is an annual it isn't worth expending much effort or money on saving it and I wouldn't use any pesticide on something I was going to eat. Especially since the larvae are inside the leave so contact products won't work and you'd have to reach for a systemic.