First, contract the Major Leagues into 24 teams total. There are entirely too many Triple-A and Double-A quality players on Major League rosters right now...some teams were eliminated in May this year, and that is a joke.
Second, establish a relegation system similar to European Soccer leagues.
The top 10 make the Major League playoffs, the bottom 4 face a separate relegation playoff.
The top teams outside of the "Major League designation" have a similar playoff and the top 2 from that playoff replace the two relegation losers in the top division for the following year. Makes the losing teams have to at least TRY not to suck the life out of their fans.
Sure, you may have the same 16 teams constantly in the Major Leagues, but the talent level would also equilibrate and instead of teams with rotations that are 2-3 quality starters deep, we would get actual staffs with 4-5 deep starters and a bull pen worth a damn.
Or we could just leave it alone and accept that this is really not much different than the league was in the "golden age of baseball" when the Yankees won seemingly every year by default from the 20's to the 60's...except for the number of teams (although an argument could be made that diversity increased the talent pool...I just think that diversity was necessary to survive for baseball - bringing in Dominicans, Cubans, Japanese and other foreign stars to fill out Major League rosters with talent.
Either way, playoff baseball is insanely intense as every game is so important compared to 1 of 162 regular season outings...sudden death baseball is tension defined and we are going to see that in the Cardinals-Dodgers game and the Yankees-Red Sox game in the next 2 days for sure.