Of the teams that had the longest World Series droughts, a group I've called "the long suffering five" (Cubs, White Sox, Red Sox, Indians, Giants), each of them seems to have a curse noted by fans over the years. The most famous was the Red Sox's "Curse of the Bambino" for selling Babe Ruth to the Yankees after the Sox won their last World Series. The Yankees went on to win 26 before the Red Sox ended their curse in 2004.
The Cubs have the "Curse of the Billy Goat," which legend has it that a fan brought a goat to a World Series game in 1945, and was told to leave because the smell was bothering other fans. He placed a curse on the Cubs after leaving. The White Sox had the "Curse of the Black Sox," which was directly due to the gambling scandal where eight White Sox players allegedly took part in throwing the 1919 World Series. The Indians have a much less known curse, the "Curse of Rocky Colavito," preventing the Tribe from winning due to the team trading Colavito for Harvey Kuenn.
The Giants had maybe the lesser known "curse" of them all - the "Curse of Coogan's Bluff." Coogan's Bluff was the place in Harlem where the Polo Grounds were located, and NY fans supposedly cursed the Giants for moving to San Francisco.
So did they have a curse? I'm not sure, and I'm not a big fan of curses, but they do make for good conversation. If they did have a curse, it's over.