To begin, the keyword in the phrase "game show" is the word "show". Some of the audience will be more interested in the game aspect of the program, but most people will watch because it is a show. Some really good games do not succeed because of poor show quality, while some shows with games that have many flaws do quite well as it comes off rather well as a show.
Being that most game shows have as its cast the host, the contestants, and occasionally the audience, the quality of the show will rest primarily with them. An audience is almost impossible to control from a selection viewpoint. The contestants are selected by the show's producers, so there's a significant amount of casting that results from that. That leaves the role of the host to complete the casting, and as the host is the principal constant, the qualities for a long-running series rest on him or her.
I would say that the key ingredient for the host himself or herself is the ability to get both the contestants and the audience to be participants in the show's theme, which depends on the type of game involved. Some game shows act as cliffhangers, with the thrill of the show resting on the dramatic aspects of the game. In this case, the host has to make the show play as a drama. In others, the game show acts as a participation game, where the viewers play along with the game, shouting answers at the TV. In those instances, the host acts like a forum leader of the like, bringing out the game aspect in the players, both on screen and off.
For both kinds mentioned above, a host is like a conductor of and orchestra, and the great conductors can turn even sub-par contestants into decent actors within the game. Usually this is done by being able to read the personalities of the contestants and interact in a way that improves their performance. Also, the conductor appears to interact with the audience, both in the studio and on the television, having it appear as he or she knows them. Probably this is best summed up as a type of charisma or charm that appropriate to the show's format.
One other major ingredient that a great game show host has is the knowledge of the game itself. All of the charisma in the world will be useless if the host doesn't appear to understand the show itself. It's not difficult for a host to be knowledgeable of the show, but this means that the host cannot just show up and walk through the game. Preparation beforehand is important, even if it is just to know how the game will evolve over the course of a given episode.
Hope that helps.
Just my semi-logical analysis of my viewing habits.