In 5 out of the last 6 presidential elections, the Republican Party has lost the popular vote. The one time it has won the popular vote was due to the war fever and panic that occurred after 9/11, and even then they only won by a meager 2%. Which is also to say that they have only really won one election in the last 20 years, and even then they barely won and it was because of an unexpected situation. And especially since this latest debacle with the government shutdown and how the press blamed them for it (personally I think both parties are at fault), I really don't think they have a chance in the next few elections either, and when this happens the party will eventually shut down.
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I don't know if it's dying, but it's definitely declining at the moment.