A sponsor was an individual or institution who supported another individual or institution financially, in legislation or in the process of applying for or being at certain institutions.
Starting in 2257, the Galactic Cultural Exchange Project sponsored the Karidian Company of Players. (TOS: "The Conscience of the King")
Spock was the sponsor of Valeris during her time at Starfleet Academy in the late 23rd century. (Star Trek VI: The Undiscovered Country)
In 2344, Chakotay asked Captain Sulu to sponsor him at the Academy. (VOY: "Tattoo")
Around the same time, the father of Martok spent much effort into finding someone to sponsor his son, so he could apply for becoming an officer in the Klingon Defense Force. (DS9: "Once More Unto the Breach")
By 2368, Pardek was known for having sponsored many reforms in the Romulan Senate. (TNG: "Unification I")
Prior to 2370, the father of Arjin had been his sponsor to become an initiate. (DS9: "Playing God")
In 2371, Chakotay promised Tuvok to be his sponsor should he ever take part in a gin rummy tournament. (VOY: "State of Flux")
In 2372, Captain Benjamin Sisko sponsored Nog's application for Starfleet Academy. (DS9: "Little Green Men") After Nog had become a cadet, he again asked Sisko to be his sponsor for joining Red Squad. (DS9: "Homefront")
In 2373, Garak practiced his "skill" in lying by asking Worf to become his sponsor and issue a letter of recommendation for his application at Starfleet Academy. (DS9: "In Purgatory's Shadow")