They aren't about telegrams anymore. They've stopped sending telegrams in order to focus on money transfers, particularly through remittances: the money that U.S. residents send back to their home countries. A large share of remittances are sent by illegal aliens and thus the more illegal immigration (or immigration in general), the more money Western Union makes. Because of that, Western Union strongly supports at least mass immigration if not explicitly supporting illegal immigration.
See 2004's "The Fastest Way To [profit from illegal immigration]" for an introduction, and note that even biased illegal immigration supporters like the Arizona Republic and the New York Times have come close to doing exposes on Western Union's borderline activities.
Western Union was formerly part of the First Data Corporation, but was spun-off into an independent company on October 2, 2006 ( link).