Compare the ability of teams to contend prior to the advent of free agency and since. Using the 16 franchises that have existed since 1903, I looked at all their records between 1920-60 (the era of Yankee dominance and before the first expansion) and again from 1976-2008, the era of free agency. That is a 41 and 33 year period.
Consider to start how often teams had a sub .500 record. There is no absolute pattern, but overall, the earlier era saw a lot more teams suffering sub .500 seasons a lot more often. For example, the Braves did it 23 times, the Browns/Orioles 31 times, the Senators 26 times, the As 28 and the Phillies 34 times from 1920-60. Those same teams did it 14, 17, 18, 16 & 15 times after 1976.
You can check how many times teams got to the post-season or won the World Series. This is less comparable as a larger percentage get to the post-season nowadays (although winning the Series is tougher). Nonetheless, 5 teams never won a WS between 1920-60; just 3 failed since 1976. Aside from the Yankees (25 times), Dodgers, Giants and Cubs who all got to the post-season 9 times, only the Cubs managed it 5 times and the other teams did it 4 or fewer. (The Tigers did it 4 times.) But since 1976, 6 teams have gotten to the post-season in double figures and no team has been there fewer than 3 times (Tigers).
You can also consider long streaks under .500. Six pre 1960 teams had a sub .500 record at least 10 straight years, including the Braves once, the Browns, Reds and Phillies twice each and the Cubs once and a second time that extended to 1963 while the As also did it once with a second time extending 15 games to 1967. The White Sox had a 9 year run. The Phillies take the prize with a 16 year run that included 4 years with winning percentages under .300.
None of those franchises except the Orioles has had a 10+ year run of .500 years although the Pirates have matched the Phillies 16 year run of futility and are poised to surpass it. Nor has any of those teams reached the same low in winning % since 1976 as they achieved prior to 1960.
Incidentally, the Yankees got to the post-season more often (61%) in the earlier period than later (55%). They also have had 5 sub .500 seasons since 1976 with only one earlier.
I know there are dozens of different ways to think about the issue, and I know that what I say here is hardly authoritative. But it at least suggests that the hysteria over Yankee spending and payrolls may be over exaggerated. Baseball has always had franchises that did poorly over long stretches and others that have been more consistently successful. Using the criteria I establish here, only the Tigers, Cubs, Indians, Pirates, Giants, Cards and Yankees have pretty clearly not done as well recently as in the past. But even in those cases, the current structure has allowed the Indians, Cubs, Pirates and Cards to appear in the post-season more often than in the past, and the Tigers to appear just one less time.