The American left today is dialectically estranged from the American mainstream and America's founding principles, although it still has the run of the education (sic) system (while missing the best tree to bark up, the philosophy one). Almost surely the very best minds among the public-intellectuals have gravitated over time to the Right (while the very best minds among the academic, non-public-intellectual population contains a mixture of increasingly-libertarian Rawls-ish liberals and the libertarians, with Tomasi representating a state-of-the-art attempt at synthesis of the two). [*]
So what sustains any impression among reasonable, decent people that the left has quality ideas and intellectual and moral credibility any longer? Some ideas:
(1) See above about the left's control of the bulk of the education (sic) institutions, a great many of them funded in whole or part by taxpayers. (Before one rationalizes like a leftist that this points to the intellectual superiority of leftism/leftists, see Nozick for a alternative explanation.) Lots of narrative-shaping going on there for many a captive audience, many of whom go out into the real world afterward rather than pursue (e.g.) state-of-the-art debates between Rawlsians and libertarians, so they don't have all that good an idea of how Rawlsians might not have any clear upper-hand despite left-wing dominance of the academy.
(2) The left is riding past successes in the areas of civil rights and LGBT+ equality. Even until very recently the American cultural Right has been hostile to progress in the latter idea, much to its motherfucking shame. The Blue States (not synonymous with the left, but a close approximation) have been making progress in the area of cannabis legalization, while outlets such as the otherwise-respectable Wall Street Journal continue to this very day to peddle bullshitted-up numbers and narratives supposedly showing how destructive cannabis and its legalization are. (The fucking least they could do is acknowledge the testimony of cannabis-users like Carl Sagan, which they don't.) What's not at all clear to me is that the Blue States are legalizing weed on the grounds of individual freedom as distinct from equitable treatment of minorities (who suffer disproportionately from the failed big-government War on Drugs) or the Blue States' greed for always-more tax revenues. And even there, the state of CA has been shitting the bed in its efforts to reconcile its big-government ethos with a free-market one.
But weed legalization is a present-day trend with the merits still to be assessed over time by the American mainstream. I don't fault the WSJ and other conservatives for having suspicions about its wisdom/merits when it's the Blue States (by and large - see Alaska for an exception) spearheading the effort.
Back to LGBT+ equality: The Civil Rights movement was over half a century ago now; the LGBT+ equality movement has been a recent thing where we can get a better sense of where the left has gone in more recent times. The usually faulty and often downright awful arguments by the cultural Right (such as illogically linking acceptance of homosexuality to the questionable aspects of the Sexual Revolution generally and their implications for (namely) widespread abandonment of the nuclear family) would lend credence to the impression that the left has the upper hand intellectually. (Note that on this and other issues the libertarians, who by default tend to end up on the "Right" side of the spectrum - especially nowadays - haven't usually been in the business of offering shitty arguments on this, cannabis, or other issues.)
(Note also that leading proponents of marriage equality include conservatives like Andrew Sullivan and Ted Olson. What would really be nice is a well-publicized debate between Olson and Princeton's Robert George, but national intellectual bankruptcy seems to preclude interest in and calls for such. No reasonable person respects the nincompoopery of the slippery-slope arguments from politicians like Santorum, after all.)
So far I've been pointing to what are broadly termed "cultural" and "personal freedoms" issues, not economic issues. This ties into the next point:
(3) The placement of figures like Rawls and Keynes somewhere near the center-left of American opinion lends credence to the notion that there are still quality ideas on "the left" generally speaking, in the smeary ways many people often think of these things (my differentiation here is one of a continuum starting with the likes of Rawls on one end and the total commies on the other, with egalitarian commitments of varying strengths uniting them under the "left" umbrella). But as with the lively Rawlsians vs. libertarians debates, there is a lively dialectic between Keynesians and more staunchly free-market positions (e.g., Hayek and Friedman, with Mises still regarded as more out on the "fringes" of free-market thought despite all his vastly learned/sage insights - a situation similar to that of Rand). With the likes of Keynes and Rawls pushing sophisticated arguments for government interventions or a large public sector generally, further-left folks like Elizabeth Warren and Obama (who's definitely well to the left ideologically despite his more pragmatic-realpolitik presidency) feel more of a license to take this further with notions about the overwhelming role of state-provided infrastructure in individual success (for which see Krauthammer's rebuttal).
Points (1-3) may not individually explain how the left maintains such a degree of (real or perceived) cultural and political respectability today, but together they may explain quite a lot. There may be other factors I can't think of right off. The values-priorities that Haidt's research points to may also explain quite a lot. Values are fundamental and prior to debates about whether this or that economic/fiscal policy is desirable; people's policy preferences are basically shaped by the values over and above whether they achieve some outcome (about which there are plenty of social-science disagreements, besides - measurement issues, replicability issues, causal understandings, etc.). This would help explain why conservatives tend to oppose efforts at drug legalization irrespective of the awful outcomes: "do we want to send the wrong message (to young people)?" (What if they applied this reasoning consistently across all policy issues? Don't lefties say that tax cuts for the wealthy "send the message" that a fifth mansion for the wealthy is preferable to ringworm medication for the destitute, whatever the actual marginal propensity to consume at the high end?) Much of political discourse comes down to signaling of virtues considered high-priority by one's favored group/party.
(There also, perceptions of value-priorities also come into play. Lots of folks - left and right - seem to get a bunch of mileage out of caricaturing libertarians as being focused on only one moral value - individual liberty - and Haidt's categorizing of libertarians in these terms plays right into this perception. Of course, libertarians are very minimalistic about the role of the state which severely constrains what they think political institutions ought to be doing to putatively achieve moral aims. I don't think you'd have much difficulty getting a shit-ton of libertarians - if there even are that many - to sign on explicitly to the principle of subsidiarity which nearly everyone ought to know more about. Anyway, caricaturing libertarians as being only freedom-focused just because their political priorities are so narrowly freedom-focused looks like a way for many non-libertarians to lazily avoid confronting the core libertarian 'self-ownership' thesis or discovering more about virtue-based libertarian thought, e.g., the Dougs or LeBar.)
Guess I might go smoke a bowl and see what else comes to mind?
[*] - Further differentiation: those I'll call the 'self-ownership' or hardcore rights-based libertarians on the one hand - Rand, Hospers and Nozick being leading examples - and the more consequences-oriented ones (Mises, Hayek, Friedman) on the other. Tomasi seeks a synthesis between these two strains of libertarian thought as well as the broader synthesis. Broadly speaking I like the idea of synthesis or dialectic, although my context (ahem) here is much more Sciabarra than Tomasi and hence probably more 'radical' in its libertarian sensibilities (within an 'Aristotelian/Randian-intellectualist' epistemological-ethical totality).