My experience, from growing basil in a climate with hot summers for many years, is that when basil bolts -- that is, puts out flower buds, you can pick those buds off for several weeks (and they come fast!) without any effect on the flavor of the leaves. I've never let a plant go into full bloom, but I have missed picking off an occasional bud, and having that bud bloom. No noticeable effect.
At that point, I figure it's getting to be pesto time, and I eventually pull the plants.
But I just ran into an expert gardener who swears that once basil starts bolting, you just hack of the top third (and make it into pesto, I guess), leaving the bottom two third of the plants in the ground. The plants will continue leafing and growing without flowering for another month or two, he says. So their life cycle isn't done when they first bolt. I intend to try that next year.