Decided to dig a little deeper.

here are all the non-championship droughts that have gone 20 years or more.
At first I thought that the Yankees would be the only franchise that has existed for at least 20 years that has not had a 20+ year losing streak. Then I recalled that they were far from the best team in their early days, and it took them 20 years of not winning the World Series before they finally got it.
As I was going, I noticed that the Blue Jays, founded in 1977 and winning their first in 1992, are indeed the only team older than 20 years to not have a 20+ year streak. Of course, if they don’t get it by 2012… they (and a few of the newer teams too, perhaps) will join the club.
Note that for the sake of ease on my part, I went with the franchise’s current team name and city.

Also, the first World Series as we know it was in 1903, so I started there for the sake of franchises like the Yankees, Phillies, Cardinals, Dodgers, Tigers, Twins, Braves, Indians, and Orioles who might otherwise be struck with longer streaks in their early years.
Chicago Cubs: 1909-present (98 straight seasons)
Chicago White Sox: 1918-2004 (87)
Boston Red Sox: 1919-2003 (85)
Philadelphia Phillies: 1903-1979 (77)
Baltimore Orioles: 1903-1965 (63)
Minnesota Twins: 1925-1986 (62)
Cleveland Indians: 1949-present (58)
San Francisco Giants: 1955-present (52)
Los Angeles Dodgers: 1903-1954 (52)
Texas Rangers: 1961-present (46)
Houston Astros: 1962-present (45)
Atlanta Braves: 1915-1956 (42)
Oakland Athletics: 1932-1971 (40)
Los Angeles Angels of Anaheim: 1961-2001 (41)
Washington Nationals: 1969-present (38)
Milwaukee Brewers: 1969-present (38)
San Diego Padres: 1969-present (38)
Atlanta Braves: 1958-1995 (38)
Pittsburg Pirates: 1926-1959 (34)
Cincinnati Reds: 1941-1974 (34)
Detroit Tigers: 1903-1934 (32)
Seattle Mariners: 1977-present (30)
Pittsburg Pirates: 1980-present (27)
Cleveland Indians: 1921-1947 (27)
Philadelphia Phillies: 1981-present (26)
Baltimore Orioles: 1984-present (23)
St. Louis Cardinals: 1903-1925 (23)
St. Louis Cardinals: 1983-2005 (23)
Detroit Tigers: 1985-present (22)
Detroit Tigers: 1946-1967 (22)
Kansas City Royals: 1986-present (21)
Minnesota Twins: 1903-1923 (21)
New York Mets: 1987-present (20)
New York Yankees: 1903-1922 (20)
Cincinnati Reds: 1920-1939 (20)
San Francisco Giants: 1934-1953 (20)
(Edited to reflect Cardinals' WS victory in 2006)