Conservatives and libertarians generally do not believe that increased inequality is a political or economic problem. To a large extent, I think that is because they fear that acknowledging the problem would require the adoption of policies they find distasteful, immoral and economically counterproductive.I see two big problems with this. First of all, there is growing evidence that inequality does cause economic problems and rather strong evidence that inequality matters a great deal for political problem.
That is, income and wealth would have to be redistributed — taken via taxation from the wealthy and given to the poor. The higher taxes will reduce the incentive to work, save and invest among the wealthy, conservatives and libertarians believe, which will reduce economic growth and lead to the expatriation of the wealthy from the United States, while fostering a culture of dependency among the poor that will reduce their incentive to better themselves and escape poverty.
The second big issue is that there is little evidence for any of the problems with redistribution mentioned (not that I don't see problems for individual redistributionist policies; I just don't believe any of the specific ones mentioned exist at any extent observable in aggregate economic evidence, while individuals may be motivated in this fashion, in aggregate, countervailing effects of the same policies overwhelm this). While there is a great deal of evidence for the wealthy shifting categories of income in response to taxes (reclassifying wages as capital income, shifting between investment categories, regular savings vs. retirement, etc.) there is little evidence of a strong income effect at high wages (though income effects can be powerful lower down the wage scale).* Expatriation pretty much doesn't happen beyond movie and sports stars (see France). And I've written a little bit about the notion of a culture of dependency, I see no evidence for it (though the idea of it is present in virtually all elites, from Confucian China to the medieval west, to the Paul Ryan; that the idea is so convenient and self-serving for the wealthy is explanation enough for its existence).
As Bruce Bartlett goes on to mention, the Founding Fathers certainly did believe there was a real threat that democracy would lead to [edit:widespread, endemic expropriation hat tip: jtf for correcting me]. However, there is no evidence they were right about this. It hasn't happened in any electoral democracy (it has occurred in a number of authoritarian states without the people's consent, but I can't think of a regime that could even generously be described as democratic that has approached widespread expropriation for any non-wingnut value of expropriation).
While there are a lot of good reasons to not like the policies proposed by Democrats, these are different reasons than those cited by Conservatives. Bruce Bartlett goes on to mention several ways of assuaging these concerns, but since the concerns are imaginary I don't see how these proposals would make policy better. As long as we're trying to solve the imaginary problems, while leaving the real problems left unsaid, we're not going to have nice things any time soon.
[Update: I also need to add that I find Bartlett's strategy suggestion unlikely to be helpful. While he's right about the split on social and economic issues I believe the general tendency is for the social issues to be weak preferences for the wealthy while economic policies are generally more strongly held. While there might be some small marginal benefit to focusing on social issues to peel away this group of the wealthy I'd bet this is an insignificantly small margin.]
*I don't see why people even find this worth arguing. With the exception of authors, movie stars, and a handful of other professions very few wealthy people face truly marginal choices in income. You can work 80 hours a week and be a millionaire, or you can make $60,000 a year for 50 hours a week (there aren't many 37.5 hour a week jobs left above median income). You don't have the option to cut your work effort by 10% because of a marginal tax increase of 10%, you face much starker choices. At the lower end of the scale, decent secure wages tend to shift people from intermittent to steady work. Basically no one will put up with shit for $20,000 a year (no matter how much we cut benefits and otherwise kick the poor), somewhere around $35,000 a year they'll keep their head down and do what they're told (even if the boss gives them shit). Hence large income effects at the bottom and small impacts at the top. I could point to research on this, but I find the intuitive logic sound enough that I don't feel the need.