Grew up in STL and started going to Busch Stadium in the early '70s: Brock, Gibson, Torre, Simmons, Hrabosky were some of the players at the time. Just a few years after their 1967 title, they had entered a swoon (no div. titles, several seasons ending 10+ games behind) which didn't end until 1982, after acquiring Ozzie Smith, Joaquin Andujar, Lonnie Smith, and Willie McGee. My family has held season tickets since I can remember.
Only the Yanks have won more WSs, and after them, only the Giants have been to the WS more than the Cardinals, which is probably why the Cards get as much attention as they do (and because they're one of the oldest teams and baseball loves traditions). Lately, they've received more press because of their farm system and the homegrown players now in the majors. The Pujols deal only burnishes the front office's image.
If you talk to the oldsters from the Rockies to the Ohio, they're Cardinals fans thanks to KMOX radio, which was the only thing on the air at night in much of the central US back in the day. I went to Kansas University, and became a basketball fan, despite having grown up with zero interest in the sport, mainly because of the Jayhawks fans I met there (and coz the team is consistently good). KU fans complain about east coast homers overlooking their team the same way STL fans do. And like patmo said, Royals fans do, too, but come on: with one winning season out of the last 20, the Royals ain't no Cardinals. And while it was hard to be in Lawrence/KC, surrounded by Royals fans watching KC win over my Cards in 1985, I still loved being in a town with fans whose team was winning. There's just something cool about it from my POV as a fan of the game.
The last 25 games have sucked: starters' ERA over 6 (Wainwright aside), only the 12th-best record in baseball after that great start, swept at home by Texas a few weeks ago after winning all but two series so far in 2013. But they're still only 2 games behind the Pirates, who are absolutely rolling: 9 Ws in their last 10 and have gone from 4 games down to 2 up on STL.
STL went thru a 30-game slump in 2011, too. They ended up winning it all. I predict the Cardinals will will right their ship, the starting pitching will come back around, and we'll see them battle PIT all the way to September, both teams winning 100+ games and perhaps meeting again in the post-season.