What does “key off” somebody’s column mean?

Solution 1:

This is a deceptively interesting question. I understood it intuitively to mean the writer was using the Krauthammer column as a topic that would set the tone for or central idea of his own piece. The usage is a familiar one, found often in journalistic pieces such as this headline.

Europe stocks gain, key off U.S. data, earnings

The phrase says a lot in a few letters, rendering it ideal for headlines.

Still, when I went to look up the term in various dictionaries, I came up empty for an exact match. I think the term must be deconstructed before it can be explained. Let's look at the preposition first. I believe it is used in the following sense (all citations from Webster's 3rd New Int'l Dictionary):

²off prep 2 a from the charge or possession of (bought it ~ a wandering peddler) (had his wallet stolen ~ him) b : from as a source of supply : at the expense of (lived ~ the country) (lived ~ his sister)

We see this usage of off in informal English (as a substitute for from), in such constructions as

I promptly bought it off him for 30k gold and gave it to Alieth.
He swaggered away through the crowd, lifted a wallet off a passing bloke ...
He always told me he took it off a dead soldier ...

So, let's suppose the off in "key off" means from. Next, let's look at the verb. This is used in the musical sense, as Affable Geek notes, but here is a dictionary citation that gives its exact definition:

²key vb 2a : to fix or determine the musical pitch of b : to regulate the musical pitch of (~ the strings).

Putting those together we get the sense of "key off" to mean (literally) "fix or determine the musical pitch from" another instrument or voice. I can attest that in orchestras, which I used to play in, this is customarily the principal oboe; in choirs it is a piano or a small reed-based device used for giving a pitch, or a choir member with perfect pitch or good relative pitch. The given pitch is a starting point for in-tune playing and harmonizing.

The literal usage has been overwhelmed by the figurative usage, though. To "key off" has come to mean to take a prompt from something else in order to produce something of one's own. The thing produced may be in concert with the prompt, may echo it or elaborate upon it, or may in some cases even take issue with it.

Finally, whatever @FumbleFingers may feel about the quality of Helperin's sentence

“On “Morning Joe,” I key off Charles Krauthammer’s column to talk about a key part of Romney’s challenge: “

it is fairly obvious that the writer, if he wasn't being merely sloppy, means to decorate his sentence with a play on two meanings of the word key. In keying off Charles Krauthammer's column, Mark Helperin aims to amplify a certain essential feeling of doubt shared by many conservatives with respect to Romney's campaign.

Solution 2:

In this instance, it is a version of "to take your cue from." Musically, you are "in the same key" or "singing from the same page." The idea is that you have a common starting point and intend to further develop what another has begun.

The idea that is being conveyed is that one commentator has read some one's column, and is using that as an inspiration or a starting point from which his own contribution is to be added.

All in all, its not the most standard construction, but it is a colloquialism I hear every now and then here in Washington, DC.