Some plants are sensitive to light changes (such as Shark Fin Melon). I think this is partially because Shark Fin Melon is supposed to rely on light changes to know when to set fruit, but even though it may wilt for a while, it recovers. I don't know of any issues with basil. However, if there are issues, I recommend just keeping the plant as strong and healthy as possible, and giving it high amounts of light, if possible.
One major issue you might encounter with bringing plants inside is that they might have such as fungal pathogens on them that weren't obvious outside, and they may kill the plant inside. That's what happened when I tried taking a cutting of an outdoor cucurbit and bringing it inside to root. I didn't have any such problems rooting cucurbit cuttings from plants I already had indoors (that germinated indoors).
Also, you'll want to be careful of pests. You might be bringing aphids (I had that happen with mint), spider mites (I'm not really sure how these got inside) and other creatures in the house with your plants, even if you don't see any on the plants outside. Pests can carry and spread plant diseases, too.