Illustrating that radical change, here's a revealing 1929 article from Time Magazine expressing some mild disapproval for what was, back then, the rare occurrence of a son who was elected to succeed his father in a Minnesota Congressional seat after the father was killed in a tragic fire (the new son-Congressman, the article noted, was "an engaging young man, thoroughly Nordic in appearance"). About this single familial succession, Time sternly intoned: "Primogeniture and hereditary public office have no place in U. S. tradition" (emphasis Greenwald's)Well, maybe. Time in 1929 did notice two Senators whose fathers had been Senators. They wiffed on a third, Guy Goff of West Virginia. Two others, Hiram Johnson and Morris Sheppard, had fathers who had been Members of the House. Another had a father-in-law who preceded him in the Senate. Then there are a couple of younger brothers...all in all, I count 11 of of the 96 Senators elected in 1929 who had prior Members of Congress (House or Senate) in their families. Not quite the 15 of 100 Greenwald reports in the current Senate, but not too far off.
(I think Greenwald's 15 may be a cheat, anyway -- he doesn't list his 15, but his links from the December 2008 post include outgoing Senators Dole, Sununu, and Clinton as well as the incoming Udalls. He also includes Olympia Snowe, whose first elective office was a state legislative seat in Maine previously held by her late husband, but I don't know of any other connection; I'd say that's pretty weak [this is wrong; update below]. OTOH, I'm not sure if he includes John McCain, who one could certainly make a case for. My search for 1929 was only for previous Members of Congress; if I had expanded it to include governors and especially state legislators, odds are I would have turned up a few more).
Just a quick look at the 1929 list reveals one possible reason for an increase, if there's been one: of the eleven nepotism Senators in 1929, all represented states that had long political histories; the newest of the ten states with nepotism Senators was West Virginia, already over sixty years old. No nepotism then from Arizona, New Mexico, the Dakotas, or the other late-19th century states -- at least not in terms of previous Members of Congress. It may take a while for states to develop old political families!
I'm going to need a lot stronger evidence to be convinced that there's a trend here.
[Update September 4: I was wrong on Olympia Snowe; after a decade in the House, but before she was a Senator, she married the then-Governor of Maine. She's not created by nepotism or family connections, but she certainly is a fair example for Greenwald to count]