It doesn't have anything to do with politics. It has to do with American Baseball.
As Wikipedia states
In Major League Baseball, the designated hitter is a hitter who does not play a position, but instead fills in the batting order for the pitcher. The DH may only be used for the pitcher (and not any other position player), as stated in Rule 5.11
As pitchers are hired for their pitching ability and not their batting ability it basically allows a team to play somebody who can bat in lieu of someone who can't. The rule is currently only used by the American League teams as the National League team owners were unable to come to an agreement over its use. The idea dates back to at least 1906 but it wasn't adopted until 1973. Like all things people still argue about it.
I'm not sure if you saw a censored version of the film but Senator Crockett's full outburst is below:
C'mon, what's wrong with you people? You're pretty loud when you want something. "Oh Senator Crockett I--I deserve better!" "Oh Senator Crockett I deserve a better life!" "Oh Senator Crockett I deserve TV dinners and T-Birds and lawns to cut and peace of mind!"
Well I got news for all of you. You're not gonna get it!
I'll tell you what yer gonna get: Porno Video with double digit inflation, and a designated hitter, and wars in countries you never even heard of!
Now that's what I call a world without magic!
His whole rant wasn't only about politics but how things change and not always how you want them to.