Aslan - Definitely the Kzinti from The Slaver Weapon TAS episode. Otherwise, TAS Caitians in appearance, Klingon in their predilections.
Darrians - Deltans without the sex pheromones and the need to take Oaths of Celibacy just to go to the shops.
Zhodani - Come on. Pre-TNG Klingons from TOS, "What is it with the Klingons? / Remember in the day, / When they looked like Puerto Ricans / And they dressed in gold lame?" ... only with psionics.
Hivers - Romulans in predilection, nothing in Star Trek remotely like them in appearance unless you count the Horta. Definitely none of the parental instincts of the Horta.
K'Kree - no analogue, in appearance or temperament.
Vargr - Orion pirates in behaviour, only without the green animal slave women (unless you include them as a conquered minor race of servitors to Vargr leaders and occasional livestock).
Droyne - In terms of psionics, think of the Talosians, Gary Mitchell, Thasians, The Old Ones from "What Are Little Girls Made Of?", the Shore Leave Planet, the Squire of Gothos, the Metrons, Landru, the Organians, Apollo, Vaal, Korob and Sylvia, Redjac, the Gamesters of Triskelion, Sargon, Thalassa and Moloch from "Return to Tomorrow," the Preservers, Gorgan, Medusans, the Melkotians, the Sahndarans ("Plato's Stepchildren"), the Vians from "The Empath," the Kalandans (and the hard light photonic holograms they left behind), Flint, the Excalbians, the Sarpeidonians and the aliens from Camus II - also Lucien from "The Magics of Megas-Tu," Q, the Douwd, and various other entities from other incarnations of Trek.
In terms of their appearance and culture, nothing remotely like them appear anywhere in the show.
Is there anything in Traveller that resembles the Gorn, I wonder?