Going into today's game the AL has a 34-21 record in 2017 interleague play. While it's still early in the season this has become standard procedure. In 20 years of interleague play the NL has won the season series only 4 times, the most recent in 2003.
This imbalance is so strong that it's highly unlikely to be a random coincidence. And what is the only significant difference between the two leagues? The designated hitter rule. While correlation and causation are two different things I don't have any other explanation.
What about the DH rule gives the AL the advantage? Many baseball people feel it's easier for NL teams to adjust to playing with the DH than for AL teams to adjust to playing without it so that would not seem to be the reason. I'm hypothesizing that the DH rule somehow results in better players populating AL rosters. It may be a difference in how teams approach player development, it may a difference in philosophy of how pitchers pitch, it may be that free agents prefer to play in the AL, it may be all of these and other reasons.
The floor is open for discussion.