The Yardbirds may not have been as influential as Buddy Holly & the Crickets, but no band is more closely connected with British and American classic rock from the late sixties until the eighties.
Three of the members of that band were the lead guitarists. First was Eric Clapton, who left because he felt the band was getting too commercial. He was replaced by Jeff Beck, and after a brief but memorable overlap, Beck left the band to be replaced by Jimmy Page.
Clapton went on to join the Paul Butterfield Blues Band, and then to form Cream, and then join the Plastic Ono Band, and then Derek and the Dominoes, before going solo. Blind Faith was in there, too. Beck formed his own band, playing with such legends as Rod Stewart and Pat Metheny. Page of course famously formed the "New Yardbirds" (Led Zeppelin) and then the Firm with Paul Rodgers. All three played with many, many others.
So literally hundreds of bands and individual artists are no more than a handful of steps from the Yardbirds. For example, Clapton played with the Beatles on
While My Guitar Gently Weeps and played with John Lennon in Plastic Ono. So the Beatles are one step from the Yardbirds. Clapton also played with Blind Faith with Steve Winwood, so Traffic is two steps from the Yardbirds. On a couple of Clapton's solo albums, his drummer was Phil Collins. Now Genesis is two steps from the Yardbirds. Rod Stewart, himself just one step from the Yardbirds, played in a fine band called Faces (two steps) with Ron Wood. Ron Wood did bloody well for himself as a Rolling Stone, so the Stones are three steps from the Yardbirds.
You can play six degrees of separation with the Yardbirds. There's no other band so interconnected with so many other bands! Check this out:
http://en.wikipedia.org/wiki/Eric_Clapton_discography . Eric Clapton played with people from Frank Zappa to Dr John to Bob Geldof. Jeff Beck has been much less active but no less diverse, and Jimmy Page's connections range from the Monkees to the Black Crowes.
So while no one can reasonably claim the Yardbirds are the most influential band, they certainly were one of the most important. Hence the honorable mention.