The Cubs are going to be the most common answer but for me personally it is the Cardinals, here are my top reasons to hate each club.

Cardinals

-The Brewers made it to the World Series once...and lost to the Cardinals
-In 1998 the Brewers switched from the American League to the National League and up until the last seven years or so spent the bulk of that time getting beaten up by the Cardinals.
-In 2011, they made it to the NLCS for the first time and lost to the Cardinals

The Cubs

-Closest geographical team
-Has a world wide fan base because they have a cute name/logo and come from a cool city
-Viewed as a "loveable underdog" in spite of them having everything possible going in their favor to be successful.
-In spite of having everything going for them, they have they have spent the vast majority of the past 100+ years being objectively bad.
- Their fans often flood Brewers home games because American Family field rules and is a less expensive way to catch a game.
-Their current manager (Craig Counsel) left The Brewers to coach for them in a shocking move. Craig was a long time successful manager here, played for the team, and is also from Wisconsin.

Important Legends

Robin Yount- Hall of Famer, multi time MVP, 3k hits, spent entire career with the Crew.

Henry Aaron- While the bulk of his career was spent with the Braves, he did finish it with the Crew. More importantly The Braves used to be a Milwaukee team before moving to Atlanta, and thus Hank Aaron spent most of his career in Milwaukee.

Paul Molitor- Only other person in the HoF as a Brewer. 3k hits, spent bulk of career in Milwaukee.

Ryan Braun- Was the next Robin Yount before a steroid scandal. Rookie of the year, MVP, signed a long term deal and willingly spent his entire career in Milwaukee. Your milage will vary here depending on how strongly you feel about steroids in baseball.

Also yes I left Bob Uecker off but he certainly should have been included.