To understand why Republicans have a "branding problem," you first need to understand how the system is rigged against conservatives.
Such is the schizophrenic dysfunction of our politics: We constantly demand "conviction" politicians who will "do what's right" and then condemn them, often in the same breath, for being unwilling to put aside their conviction and their sense of what's right.
But such condemnation does not fall equally on conservatives and progressives alike. For the progressive's principle is, at its core, more. Do more. Spend more. Spend more doing more. Any compromise of progressive principle in this regard is seen as "pragmatic." Hence, the progressive's heart is always in the right place.
The conservative, however, who says the federal government is not the right tool to fix the problem at hand, or that it is not Washington's job to fix said problem, or that such a problem is itself not fixable and taking money from taxpayers to try is despotic folly: This conservative's heart is never in the right place.
In other words, the progressive wins entirely on the principled question of direction. The conservative (or libertarian) loses entirely on principle but gets concessions on how fast we'll go in the wrong direction. The progressive says, "Let's move to Mars." The conservative says, "Earth is fine." They compromise by moving to the moon. And, before the first lunar dawn, the progressives start agitating about how Mars would be so much better.
When the classical liberal philosopher Friedrich Hayek famously said that he couldn't call himself a conservative because "It has ... invariably been the fate of conservatism to be dragged along a path not of its own choosing," he had this dynamic in mind, and you can see it on full display as progressives respond to the unfolding disaster of ObamaCare by arguing for a single-payer system.
This gets to the heart of why the Republican "brand" is in such terrible shape. Over the 20th century, progressives erected a system and culture where the government in Washington is the agency of first and last resort for all of our problems. When government is expected to say yes to everything, electing the Party of No makes as much sense as hiring a priest to run a brothel.
So what is the answer? Many conservatives argue that what the GOP needs to do is start saying "Yes" to things. This was the idea behind George W. Bush's compassionate conservatism. Americans want an activist government, so conservatives should find things they can be activist about, too. If the government is going to meddle, it might as well meddle in conservative ways.