Since the pest is entering through the calyx, it likely is the tobacco budworm, which is known to feed on petunia buds, flowers and seeds. If it is the budworm, or any other kind of caterpillar, you should be seeing black frass (unless rain has fallen and washed it away). Otherwise it might be some kind of weevil.

With the basil seeds, it is your choice. Basil produces and ripens seeds differently than lots of other plants and can have mature black seeds ready for harvest while still developing younger seeds, which are green. So, if you want to collect as many seeds as possible, you can go from flower to flower harvesting the dried flowers that have black seeds inside them while leaving the dried flowers that have seeds which are still green on thr same flower stalk. Or, when the whole flower stem is brown and dry, you can clip each stem off the plant. An easy way to "catch" the seeds after that is to hang the stems upside down inside a paper bag indoors in a cool, dry location. As the seeds dry and reach a certain maturity level, they'll fall out of the flowers and land in the brown paper bag. When all the seeds have shed from the flower heads, you can toss the stems and keep the seeds.

I use a simpler method myself. I clip off the brown, spent flowers and toss them everywhere in the garden as soon as I cut them. I drop them right on top of the mulch. I drop them in the pathways, I throw some of them onto the compost pile or on the ground around its edges. If I specifically want to have basil in a different fenced-in garden area the following year, I carry some brown flower stems to that garden plot and toss them on the ground in that garden plot. The following year in mid-spring I'll have little basil plants popping up all over the place. When they are about an inch tall and have a couple of true leaves, I transplant them to whichever place in the garden I want them to grow that year. Easy-peasy, no drying and storing seed and no indoors seed-starting under lights required.

Of course it rained yesterday, so no frass. I saw no damage to the flowers, just a few holes in leaves. I didn't see any critters on them. So sad that even my petunias aren't safe.

I am saving seeds to have for seed exchanges. I was saving Tulsi seed, which grows like basil. I am growing Genovese basil, lime basi and a lenon basil which, sadly is a hybrid. I don't know how easily they would cross, but the Tulsi should be safe to save. I have a seedling of Blue Spice basil which is supposed to have a vanilla scent. I am kind of a herb junkie. I don't necessarily USE them, I just like to grow them, like flowers.