Colvins in my CSA basket. Pesto, here we come!
Simply cut stalks right above a pair of leaves. Sharp scissors are helpful, but I've used sharp fingernails in a pinch. Then, plunk the stalks into a cup of water. Any leaves that are below the water line will turn back. It's best to have enough water to cover the bottom stem ends by an inch or so and just replenish. If you're getting basil from somewhere like a CSA basket or farmer's market, you may want to give the stems a fresh cut.
Do not store in the fridge. I store mine on my kitchen windowsill which gets filtered sunlight. Any similar spot should work fine for you.
You can also pull the leaves off and freeze them on a cookie sheet. When frozen, toss them all in a bag. They'll turn black, but the flavor will be far superior to the dried stuff and will work well in cooked dishes (like spaghetti sauce or tomato basil soup) in the months to come.