Adam Liptak and Allison Kopicki recently had an interesting analysis of public opinion on the Supreme Court. The public reaction to the health-care ruling, NFIB v. Sebelius, shows that the public is closely divided, with 46 percent supporting the decision. As expected, NFIB has made Democrats like the Court more and Republicans like the Court less. Putting the public reaction to NFIB in context, Liptak and Kopicki compare the public reaction to the case with other high profile cases. The data is somewhat sobering for advocates of civil rights and liberties.
Before getting to the bad news, it's worth noting that despite the cottage industry of pundits dedicated to the idea that support for women's reproductive rights is undermining the Democratic Party, Roe v. Wade is an exception to the general rule of liberal civil-liberties decisions being unpopular. Roe was popular the day it was decided and its popularity has, if anything, increased over time. Conversely, Gonzales v. Carhart, which upheld the federal ban on "partial birth" abortion, is not especially popular, which indicates that the Court would not face a significant backlash if it were to strike down the new wave of regulations designed to make it impossible for abortion clinics to operate.
Outside of reproductive freedom, however, pubic support for civil liberties and civil rights is more limited. Liberal Establishment Clause rulings in 1966 and 2000 were extremely unpopular, as was the 1989 ruling that flag-burning was protected by the First Amendment. Even the relatively mild rights protections offered to arbitrarily detained "enemy combatants" in Boumediene v. Bush were similarly unpopular. The public does support liberals on the campaign-finance decisions, but since these are cases in which most liberals favor a narrower reading of the First Amendment than conservatives, these are the exceptions that prove the rule. Given that the Roberts Court will have the opportunity in the next term to rule all state affirmative action unconstitutional, perhaps the most disturbing data concerns the precedents in that field. Parents Involved, which perversely read Brown v. Board of Education as preventing school districts from pursuing modest means of integrating, is the single most popular of the landmark decisions compared in the article. Conversely, 1987 and 2003 rulings that read civil-rights law and the Constitution, respectively, in a way that permitted affirmative action in some circumstances were extremely unpopular.
The unpopularity of affirmative action does not guarantee that the Roberts Court will rule it unconstitutional in all cases; the Court, as the sum of these decisions also makes clear, does have some autonomy to protect the rights of minorities even if they're not supported by a broad constituency. But the Roberts Court wouldn't face a backlash if it did so, which certainly doesn't help.