Comparing players from different eras always invites an increased degree of speculative thought, but I tend to agree with Nicklaus here. Year in, year out, Nicklaus had more competition from a number of players. More players in that era won tournaments than seem to win nowadays.
One of the most telling statistics in modern golf is that there are only five golfers under the age of thirty currently on the tour (excepting Tiger who won plenty in his 20's) who have won two or more tournaments during their 20's. Sergio and Adam Scott have each won three and Baddeley, Trahan, O'Hair, Holmes and Howell have won two each.
The PGA Tour is littered with players who get their card every year and never get a sniff of a win, but they still get endorsements and make enough money to keep at it. (The one exception to the endorsements seems to be Briny Baird, who was sporting a PJ Chang's hat and a shirt devoid of any logos during the New Orleans stop.)
Returning to topic, there were more great players in Jack's era, including Trevino, Palmer, Weiskopf, Watson, Floyd, just to name a quick few. And they were all great during a long time when Nicklaus was predominant. Nowadays, we have much less competition. We either have tournaments in which Tiger competes (he kicks butt and very few challenge him) or tournaments without Tiger (where we have a bevy of one or two-hit wonders who fade into the background only to be replaced by another blip on the golf radar screen).
Jack finished twice dozens of times. When he did, he usually lost to a great player, not a journeyman like Duffy Waldorf or a kid like Johnson Wagner. It may be nostalgic, but I think Jack is right on this one.