I'm across the bridge from you in NJ :)

For basil I think in these terms: we are told never to put it in the fridge b/c it will turn the leaves black. So I won't even think of putting it outside to harden off until the outside temp is consistently 65-ish (of course bringing it in at night)

I am impatient to get these plants going this year! I think I will give myself a target date of Mother's Day to get the impatiens into their containers, my thyme will be able to stay outside full-time by then, and probably even my seedlings.

I may take my basil inside at night until maybe a week after Mother's Day, just to be safe.

Hardening off is ok, but if the temps do drop too low overnight, it not be good for the plants. MY small greenhouse can get to over 100 degrees in the middle of the day, but even just 3 days ago, we had a light frost overnight. Your in Z7, so you should not be seeing any frosts. Even if the leaves do get a little dark, the plant does grow many more. Here, I have to wait until almost the end of May

Does anyone know if basil could be grown indoors?
Thank you.

Basil grown indoors will usually give you spindly plants. They love plenty of sun, and because its indoors, it will need plenty too. Using grow lights for 16 hours per day can help a little, as can some small amounts of nitrigen based fertilizer, but beyond that, the true flavor, color, and vigor will not be the same as plants grown outside.