Hmm.. I think I have a 3-Way tie.. All 3 of the 3D Mario games: Super Mario 64, Super Mario Sunshine, and Super Mario Galaxy.
I know all Mario 64 did was show us a cryptic letter about the Princess inviting Mario to the castle for some cake, but I want to know how many people jumped (literally jumped) when you walked into the castle, only to hear Bowser's evil cackle coming from the walls. Eerie!.. Otherwise, it set the game up well, and even had throw backs to the 2D games (letters from the Princess, items from Toad, cake XD
Mario Sunshine had a good intro, setting up the story even better than Mario 64 did. The in-flight movie showing a very Mario-esque character ruining the Island. The only gripe that I have with this game (aside from the fact that it wasn't a very good Mario game) is that if anybody had just listened to the princess, her capture in this game could have been prevented.
The intro to Mario Galaxy is awesome. It ties together many of the themes from previous Mario games and sets up the story extremely well, making you want to play the game so that you can find out more. It even shows what happens when Bowser invades the Kingdom. The only complaint I have about this one (a minor complaint) is: How did Bowser get aliens to side with him? There are several alien ships present at the time the Princess (and the castle) were captured.
If I have to pick just one, it would be the Galaxy intro, with Sunshine and 64 falling into second and third, respectivley. But honestly, I had a hard time picking just from the 3 as so many Mario games have good intros (or games that feature Mario, like the SSB series).