Q: How can I perk up fresh basil or other delicate greens that have gone limp?
If they are just a little limp you can try recutting the stems (the shorter the better) and placing them in a small container of water like flowers. I have read that 110 degree(f) water maximizes the uptake into the stems. Actually keeping your fresh herbs that way from the begining (including recutting the stems) will get you better shelf life.
If that doesn't work think of using them finely chopped or in a pesto!
Or, you could try putting them in a large bowl or other container filled with very cold water (large enough so there is a lot of room around them) and then put that in the fridge for a few hours. That works well for arugula, spinach and most lettuces, if they have not gotten too limp.
You could try putting the stems in a glass of water, placing them on the counter, and covering them with a plastic bag. It creates a little greenhouse of humidity for them so they're getting moisture through the stem and the leaf. I honestly haven't tried it, but this is an outstanding way to keep them fresh as new when you've first purchased them and may be worth a try to perk them up as well.
I agree with the stems in water approach. I typically treat leafy herbs like cut flowers. I have some fresh cut sage on my counter in a vase like container and the leaves seem to be holding up just fine.
A little late, but: put them in a glass of fresh water with a little sugar in it. Its a trick I learned as a waitress to keep the table flowers perky.
Plant it and simply pick as you need it.