Which Chicago team has the largest fanbase,,the Cubs or the White Sox?

I would say the Cubs largely because of two factors....

First, the Cubs fanbase is traditionally more affluent than the White Sox, owing to their respective locations in the city (Wrigley Field being on the fashionable North Side while Comiskey Park in the heart of the blue-collar South Side).  This means the Cubs fans tend to travel, relocate and retire out of the area to a greater degree.  It's no accident that there are sizeable Cub fan bases in warm weather retirement destinations like Arizona and Florida.

Second were the so-called "WGN years."  When the Tribune Co. owned both the Cubs and the TV network, they began broadcasting every game.  When WGN went national as a cable channel, they brought Cub games with them to baseball-starved fans across the country.

Not only did this spread interest in the Cubs, but also made Wrigley Field traditions like throwing HR balls back from the stands and making the 7th Inning Stretch an "event" commonplace in other stadiums as well.

Cubs.  No contest.