Since its inception the American League has dominated in season inter-league play. Up through the 2008 season the American league had a record of 1536 wins and 1420 losses.
The American League has also dominated the All-Star game having not lost since 1996. However, this isn't a fair measure of which league is better because a league can easily have the best group of 25-35 stars but still be the weaker league.
A more appropriate measure is success in the playoffs. Since the American League adopted the designated hitter in 1973 they have won 20 of the 35 World Series played. In the 17 World Series' played since 1991 the American League has won 11 of them.
The American League has dominated almost every aspect of baseball success since installing the designated hitter.
The reason for this dominance is fairly simple. Half of the time when playing a National League club the American League has one extra starting quality player.
When general manager is putting his team together prior to the season he is looking to allocate the bulk of his teams funds to starting pitching and starting quality everyday offensive players. Certainly some of the teams funds must be spent on bench players and depth, as they play a significant role on any championship run.
However, they are paid far less than starting players.
In both the American League and National League General Managers will spend a large portion of team funds on pitching. Where the difference lies is on offense.
And American League club will allocate funds for nine starting quality offensive players. Whereas a National League club will spend money on only eight starting quality offensive players, probably spending slightly more their bench.
Now, when in a National League ballpark both teams will have to bat their pitchers. There is no discernable difference between AL and NL pitchers. They are all equally awful. Free agency and pitching-centric routines strip pitchers of any hitting talents they may have leaving all if not most completely useless.
One might argue that in an NL park the AL team is at a disadvantage because they might have to play their DH on defense because they can't afford to lose their offense (see David Ortiz for the Red Sox teams).
This is certainly reasonable and likely does slightly hurt some teams defensively. However, national league clubs also have defensively challenged players who they need because of offense and play them just the same. The difference is probably negligible in most situations.
Everything changes when you are in an AL park. Here, the teams are permitted to use a DH. The AL team likely has a quality DH who they are paying well to be a quality DH. He was part of their plans the entire time and he is likely a starting quality offensive player.
However, in an AL park most NL teams don't have a starting quality offensive player to slip into that DH role. Most NL teams do one of two things. Either start their best offensive bench player at DH or put their weakest defensive starter into the DH role and start a good defensive player from the bench in his place on the field.
In both situations the extra player who plays because of the DH rule (I will call him the 9th man) is a player who when signed was considered a bench player and paid as one. He is not a starting quality offensive player.
So in an AL park during inter league play or the world series the AL is starting nine starting quality offensive players whereas the NL is starting eight starting quality offensive players and one bench quality offensive player.
To illustrate my point I will compare the "9th man" of each current division leader as well as the New York, Los Angeles, and Chicago teams and the Cardinals to even the number of teams. The definition of the 9th man is the player you gain when you play in an American League ballpark.
For AL clubs the 9th man is not necessarily the DH. The 9th man for an AL club is the guy who wouldn't start if the game was in an NL ballpark. The same is true for the NL. The DH for an NL club is not necessarily the 9th man. For an NL club the 9th player is the player who would not start in an NL park.
Here is the list of likely 9th men for the American League:
American League 9th men tend to be older players who are defensively weak. They tend to be power guys with large OPS. They tend to play relatively important roles in the offensive lineup, though aren't one of the major cogs...otherwise the teams would find a place in the lineup for them.
Here is the list of 9th men for the National League:
Matt Stairs/John Mayberry (Phillies)
Craig Counsell (Brewers)
Nick Stavinoha/Ryan Ludwich/Chris Duncan(Cardinals)
Mark Loretta (Dodgers)
Jeremy Reed/Fernando Martinez(Mets)
Milton Bradley/Micah Hoffpauir(Cubs)
National League 9th men tend to be either professional pinch hitters, platoon players or defensive replacements. They are overwhelmingly not every day players. Most NL clubs don't have a single set 9th man, evidencing a lack of a quality player to fill that role.
It is clear from the list above (purposefully made up of playoff contenders and big money teams) that when in an American League park the AL team has a significant offensive advantage.
Perhaps one could argue it evens out because in the National league parks the AL clubs lose these players. While that is certainly true when you compare the NL park lineups to their usual lineups it does not mean they don't have an overall advantage.
Few of these players are the major cog in their teams lineup. If they were the team would find a place for them in an NL park (see Ortiz as 1B the past few years).
One could also argue in NL parks AL clubs lose big on defense because some are forced to play a DH in the field. However, NL clubs also have weak defensive players who play every day. In the NL you have to sacrifice defense for offense because the NL doesn't have the DH to hide a weak defensive player (see Gary Sheffield/Adam Dunn/Manny Ramirez).
Because of this I am willing to say the loss of a relatively strong player in an NL park, mitigated by the fact that there are 8 other starting quality offensive players, is less effectual than the gain made by an AL club when they start their 9th man against an NL 9th man, mitigated by the gain in defense.
In summation, the DH and the role it plays on AL clubs combined with how the AL general managers stock their club with offensive players based on that rule give American League clubs a distinct advantage in both inter-league play and the World Series when games are played in an American League Ballpark.
This advantage outweighs any advantage National League clubs attain defensively when games are played in a National League club.
To create an even playing field baseball should either install the DH in both leagues or remove the DH from both leagues.
I am a baseball purist to a certain extent. I believe in the old sandlot rule that you don't get to hit unless you play the field. I would remove the DH entirely. I know full well the MLBPA will never less this happen, so it is likely to go the other way if anything ever change...unfortunately.