This was a humorous short story that I most likely read in one of the "big name" US science fiction magazines sometime in the 1980s.
The story concerns the arrival of an alien starship bearing beings who claim to be the ambassadors of a galactic union of some kind. The initial enthusiasm for their arrival is quickly tempered by the fact that they are essentially low-rank and low-skilled bureaucrats sent to this backwater planet on the rim, not a shipload of extraterrestrial scientists and philosophers.
The protagonist of the story is someone (a humanities grad student?) who joins the human staff of the embassy and sticks around after all the hoopla fades because it's a decent job. He enjoys his duties except for sorting the mail, as many people send things (gifts, pleas, rants, etc.) to the embassy without solicitation. I think he does this once a week or once a month, and he calls it "Trash Day."
Somehow the lead alien ambassador gets interested in horse racing. I think that the alien is able to work out the winning horse every time through mathematics somehow, but that's not the point of the story. After a while, more aliens arrive at the embassy, and the ambassador -- who is normally very open with the protagonist -- starts being secretive. I think the protagonist is hauled before a military tribunal that is worried about spy satellites recently launched by the aliens, which are considered a possible precursor for invasion. It later turns out that the ambassador has the satellites pointed at race tracks all over the world and that horse racing has become a galactic fad of giant proportions.
The ambassador is making money out of this, but things have gotten out of hand. The climax involves a major horse race (Kentucky Derby?) that has to be won by a certain horse in order to prevent a major economic catastrophe across the galaxy. I don't recall the details of the end, but I think it's a happy ending in that the right horse wins and the crisis is avoided.
Update: I remembered a little bit more about the story. The horse that was supposed to win the race to prevent the problem was the worst horse on the field, with long shot odds. The US government had to bribe the jockeys of the various other horses to get them to intentionally lose the race. The jockeys had to take stronger and stronger measures to make it look like the designated horse's win was legitimate. In the end, I think a massive fistfight between the jockeys results in the disqualification of every horse except the chosen one.