I wrap the bottoms in a wet paper towel and stick in a glass, dampening as needed. I think it should be fine.

sometimes if I have just leaves without stems to put in water, I put them in a plastic bag in my wine fridge at 58 degrees or so. The leaves don't seem to turn black at that temp.

if you cut the stems and put them in a glass of water (like cut flowers) they'll stay fresh a couple of days - refridgerator is bad for basil leaves!