Cut the whole stalk. The stems (edible part) are attached at the base, and the stem is only for the flower. My rhubarb flowered last year, and I was fascinated by the development from the curious shiny "bud" to the little green brainlike thing that emerged to the tall flowering stalk.

I was trying to give the plant away last year, so I wasn't concerned about lower production. But now I have more rhubarb, not less. A couple people dug up root chunks, and that must have encouraged the plant to produce more crowns. I now have 4 root crowns!