The best example I'm aware of of a campaign using Silver's numbers to validate their own polling is described in The Center Holds: Obama and his Enemies by journalist Jonathan Alter. It relates to Obama's 2012 re-election campaign, at the beginning of which, the New York Times Magazine published Silver's infamous Is Obama Toast? article which described the President as "a slight but not overwhelming underdog".
This article seems to have served as a reality check for Obama's campaign managers. Alter describes the reactions of Jim Messina & David Axelrod:
In Chicago Jim Messina practically choked on his breakfast. Obama was
trailing the front-runner, Romney, by 3 or 4 points in national polls
but looking better in battleground states. Messina fired off a
rebuttal to the Times Magazine and vowed that after winning the
election, his trophy would be a framed copy of the “17 percent chance”
cover with the president’s signature. David Axelrod happened to be in
St. Louis when he read Silver’s story. He noted that in September
Silver had predicted that the St. Louis Cardinals had only a 1 percent
chance of winning the World Series. In October, one strike away from
elimination, they won it in seven.
While this example doesn't mention Silver's analysis informing specific campaign decisions, it seems likely that this event almost a year before polling day had an effect on Messina in particular, possibly influencing his choices over the campaign. Towards the end of the book, Alter mentions:
After the election Jim Messina had one favor to ask the president,
which was easily granted. In mid-November he walked into the Oval
Office, where Obama presented him with a signed and framed cover of
the New York Times Magazine with Nate Silver’s “Is Obama Toast?”
The effect of this analysis on at least the emotions behind the campaign is backed up by Dan Balz in his book Collision 2012: The Future of Election Politics in a Divided America, in which he quotes Axelrod after Obama's victory speech:
“I felt the weight of the burden on [Obama] more than I felt the elation
four years ago,” he said. “I know he believes this was a more
satisfying win even than that one because of all the obstacles. A year
ago the sainted Nate Silver was writing a magazine piece the headline
of which was, ‘Is Obama Toast?’ To be written off and degraded and all
that and come back and win . . .” He didn’t need to finish the
On a more concrete basis, it seems Silver's numbers were also used to validate private polling after Obama's less than stellar performance in the first presidential debate on October 3rd. Although the debate went poorly for Obama, his campaign was hopeful that it wouldn't swing votes away from Obama, and the effect would be limited to Romney winning back votes he had lost since the Democratic Convention. Alter describes the campaign being advised to "do the hardest thing in a tough situation: nothing". This decision was later validated by Silver's model.
When public polls and Nate Silver’s model showed Obama in trouble,
Chicago continued to fret. Then, less than a week after Denver, David
Shor, the former child prodigy in the Cave, dove into the latest
numbers and told Kriegel, “Tomorrow, Nate Silver goes up.” Sure
enough, the New York Times polling analyst, a security blanket for
millions of Obama supporters, reported the next day what Chicago had
already heard from the Cave and from Benenson: No big change.
The Golden Reports from Analytics found the race stable, with one
exception: Numbers coming out of the Green Bay area showed Romney
expanding his lead there from 2 points to between 6 and 9. This didn’t
sound right, but the Cave’s models couldn’t be dismissed. The campaign
bought more ad time and eventually sent both Obama and Bill Clinton
into the region, even though it may not have been necessary.