MLB has only had a wild card since 1995 and two wild cards since 2012. In the days of 1 wild card they played a 3 game series against the team with the best record. How have they fared?
1995 Yankees and Rockies were wild cards and both lost their opening series.
1996 Orioles and Dodgers were wild cards. Orioles won their series then lost in ALCS. Dodgers lost opening series.
1997 Yankees and Marlins were wild cards. Yankees lost opening series, Marlins won the World Series.
1998 Red Sox and Cubs were wild cards and both lost their series.
1999 Red Sox and Mets were wild cards, both won opening series and lost championship series
2000 Mariners and Mets were wild cards, both won opening series, Mariners lost championship series, Mets lost World Series
2001 Athletics and Cardinals were wild cards and both lost opening series
2002 Angels and Giants were wild cards and Angels beat Giants in World Series
2003 Red Sox and Marlins were wild cards, Red Sox won opening series and lost championship series, Marlins won the World Series.
2004 Red Sox and Astros were wild cards, Astros won opening series and lost championship series, Red Sox won World Series
2005 Red Sox and Astros were wild cards, Red Sox lost opening series, Astros lost in World Series
2006 Tigers and Dodgers were cards, Dodgers lost opening series, Tigers lost World Series
2007 Yankees and Rockies were wild cards, Yankees lost opening series, Rockies lost in World Series
2008 Red Sox and Brewers were wild cards, Red Sox won opening series and lost championship series, Brewers lost opening series
2009 Red Sox and Rockies were wild cards, both lost opening series
2010 Yankees and Braves were the wild cards, Yankees won opening series and lost championship series, Braves lost opening series
2011 Rays and Cardinals were wild cards, Rays lost opening series, Cardinals won World Series
2012 Orioles, Rangers, Cardinals, and Braves were the wild cards, Orioles won wild card game then opening series. Cardinals won wild card game and opening series, lost in championship series
2013 Rays, Indians, Pirates, and Reds were wild cards. Rays and Pirates won wild card games then both lost opening series.
Add it all up and there have been 42 wild card teams over 19 seasons. Five of them won the World Series, which is just over 25% of the time. Another 5 made it to the World series and lost, so about half of the seasons a wild card team has made it to the World Series, the 2002 World series was 2 wild card teams.
In other words, the wild card is a more difficult task but it is not the kiss of death.