From the Super Mario Wiki
Santa Claus, (usually simply Santa, alternatively known as Nick) is a jolly man who gives presents to people on Christmas Eve. He has made a few appearances in the Mario series, and has received many allusions in games. Santa Claus lives in a home near his workshop at the North Pole, where his elves make toys. Santa Claus also rides around in a reindeer-drawn sleigh. Santa Claus has been seen on both Earth and on Mushroom World.
In games, Santa Claus has only appeared in Wrecking Crew, as one of the prizes similar to the Golden Hammer, though he did appear in The Super Mario Bros. Super Show! and the Club Nintendo comic series.
Smithy bears a resemblance to Santa, and Nintendo Power even stated in the Official Super Mario RPG: Legend of the Seven Stars Player's Guide that Smithy's first form resembled a "diabolical Santa Claus."
The Super Mario Bros. Super Show!
In "Koopa Klaus", King Koopa had frozen Santa's workshop, and dressed up as Santa himself, calling himself "Koopa Klaus". Koopa Klaus then kidnapped Santa, and attempted to ruin Christmas, but Mario and his friends rescued Santa. Santa was happy to be saved, but remarked that he couldn't make any toy deliveries since his workshop was still frozen. Princess Peach urges Toad to cheer Santa up by giving him his beloved snowboard for Christmas which the Mushroom Retainer does with no regrets. Upon offering his snowboard, Santa becomes so touched by the kindness expressed by Toad that the warmth of holiday cheer melted Santa's workshop. Santa then went to deliver presents and let Mario and friends ride with him.
"Santa Claus Is Coming to Flatbush"
In "Santa Claus Is Coming to Flatbush", he turned up at Mario Brothers Plumbing on Christmas Eve, because someone had stolen his "transportation". As he used the name Nick in this episode, Mario and Luigi never concluded that he was Santa Claus, despite obvious, subliminal messages throughout his stay. Worn out and tired, Mario and Luigi felt obligated to take in the stranger on Christmas Eve, and treated him to both their dinners, despite having very little to eat.
After the CooKoo Bird announced the midnight hour on Christmas Day, Mario and Luigi rushed to open their presents, but they looked back at Nick. Feeling sorry for Nick, they offered their presents to him, who was grateful to be receiving presents on Christmas for once.
In the morning, Mario and Luigi woke up to see that although Nick had left during the night, he left behind presents for the brothers, along with a note stating that the cops had found his transportations. Mario and Luigi then saw Santa Claus ride by the basement window on his sleigh, but had serious doubts that their visitor was actually Santa Claus.
Super Mario World television series
"The Night Before Cave Christmas"
Although Santa Claus himself never appeared on the Super Mario World animated series, Mario did dress up as him to help emulate the spirit of Christmas for the Cave People during the episode "The Night Before Cave Christmas". Dressed as Santa, Mario eventually sets out to deliver toys to all of the Cave Children in a Winged Yoshi-drawn sleigh.
"Super Mario: Die Bescherung"
In the 1992 Club Nintendo comic "Super Mario: Die Bescherung", Mario and his friends throw a Christmas party. When Mario leaves the house and shortly afterwards a man dressed as Santa Claus, but otherwise resembling Mario, enters the house, most of the guests are expecting that it's Mario dressed as Santa Claus. However, in this moment Mario joins the party again, astonishing the skeptical guests.
Three years later, a man looking almost exactly as Santa Claus in "Die Bescherung", was seen in the comic "Warios Weihnachtsmärchen". However, Wario calls him Mario and he does not correct him, so in this comic it was just Mario dressed as Santa Claus.
"Donkey Kong in: Banana Day 24"
A differently-looking Santa Claus also appeared at the end of the story "Donkey Kong in: Banana Day 24", joining the Kongs' Christmas party and bringing gifts. He also mentions that he is surprised to see Earth back in its original position (in the story, aliens accidentally pulled the planet away from the sun, and the Kongs convinced them to return it).
Super Mario 64
Santa Claus was mentioned very briefly in Super Mario 64. His name was stated in the sign near the cabin at the of Cool Cool Mountain, where a sign said "Santa Claus isn't the only one who can go down the chimney!"
Paper Mario series
In Paper Mario, if Mario chooses to sleep in the Toad House in Shiver City, he will occasionally wake-up to find several Mushrooms and an Ultra Shroom in the Toad House; the Toad caretaker of the Toad House also implies that it may have been Santa Claus who left Mario these gifts.
Super Paper Mario
Mario & Luigi series
Mario & Luigi: Superstar Saga
Mario & Luigi: Partners in Time
The Hollijolli Mayor of Mario & Luigi: Partners in Time is a reference to a Santa for the Toads, being the mayor of the winter-themed Hollijolli Village and even getting stuck in a chimney while wearing a red costume with white beard and having a Toad-like appearance.
Mario Party 7
In Mario Party 7, there is a Bowser Time event exclusive to Neon Heights wherein Bowser will refer to himself as "Santa Bowser", before stealing either ten coins from every player, or a Star from the player in the lead. Bowser will then order Koopa Kid to add the loot to the Treasure Chest on the board that already contains either the Star or twenty coins, noting that the player who claims the prize is getting a bonus out of it.
Mario & Sonic at the Olympic Winter Games
Santa Claus appears as a VIP member in Mario & Sonic at the Olympic Winter Games during the month of December. He gives out 300 Star Tokens rather than 100 like the other VIP members. This is the only time Santa Claus himself appears in a Mario game.