For one thing, I don't see it happening. From my perspective, nearly everyone I meet is middle class; no way is the middle class shrinking. I figure if you have a decent place to live and food on the table without having to work 80 hour weeks in terrible conditions, you're middle class, unless you don't have to work at all, in which case you're rich. It may be true that income growth has stagnated for all except the superrich, but that just makes them richer, it doesn't make the rest of us poorer.
This graph sort of says what I mean:
The poorest people in the U.S. are still richer than the richest in India. I'd say that pretty much makes them middle class, or at the very least pushes the status of the American middle class wayyyyyy way down on my list of world priorities. Isn't it more important that everyone has enough to survive than that everyone survives at the same level?
It seems like most of the concern about income inequality comes from resentment towards the superrich. I could care less if there are twice as many Bill Gates's running around than ten years ago. Even if they don't "deserve" all of their wealth, whatever the heck that means, they're not stealing anything from me, so who cares? Most of the superrich create a lot of value for the world. How that value should be split up between them personally and everyone else just doesn't seem like a big deal to me. As long as it's not a transfer of wealth from the masses to the superrich, we're still better off for their existence, and there are more important things to worry about...
I guess you could say I have a very "American" view of poverty, as Daniel Hamermesh succinctly describes it. In the European measure of poverty, the bottom X percent of people are always defined as poor. In the U.S., you're not counted as poor unless you can't afford a basic subsistence existence. That makes so much more sense to me that I can't even imagine how the alternate definition would take hold.
Playing the devil's advocate though, I know that happiness does depend somewhat on relative standing. In particular, we prefer to be friends with people at the same socioeconomic level. It's hard to hang out together when one person can barely afford McDonalds and the other wants to get bottle service at a ritzy Manhattan nightclub. If income inequality were so suddenly distorted that social groups are ripped apart, that would be unfortunate. (Slow changes are an inevitable part of life that aren't worth fighting.) But that certainly is not the case right now.
Additionally, simple jealousy of others, regardless of whether they are close friends, is also a tangible influence on happiness. Part of me wants to say to that, get over it. Focus on yourself and appreciate what you have. More importantly, these subjective, unverifiable, and unnecessary, components of utility can almost never be a basis of policy because tangible physical harm must be done to someone else to avoid them, and tangible physical harm always trumps hurt feelings. (That seems obvious but there are so many instances where that principle is rejected, because the people with hurt feelings fight hard for their side, usually with moralizing logic that defies the reality that they just don't want their sensibilities offended.)
On the other hand, these subjective emotional components of utility are real even if they're hard to quantify and not inevitable, and therefore they have real consequences. If society consists of starving masses and a few elites, even if the elites aren't actively repressing the masses, that can induce enough resentment to spur unrest or uprising. This is also a situation so completely different from America today that it's not a cause for concern. No one is being held back by the superrich in this country. Many are being given jobs and opportunities much better than they could hope for if the superrich all left the country or lost their fortunes. Even poor Americans have too much to lose to take a gamble on a revolution. In that situation, any residual jealousy is just not enough to motivate a radical change in policy that might rectify the ginni coefficient in exchange for all the collateral damage those policies would inflict.
Much better to advocate for a culture that is less obsessed with comparing numbers on paychecks, to kill the jealousy at the source with no collateral damage at all.