For now, a few brief thoughts:
The Court has a whopping 62% approval rating - a level Trump or Congress would kill for. For the moment, the Kavanaugh confirmation has joined a long series of other events that partisans on one side or the other thought would seriously damage the Court's legitimacy, but ended up largely failing to do so (e.g.Roe v. Wade, Bush v. Gore, Citizens United, The Bork and Thomas confirmation battles, etc.).
However, the growing partisan divide over the Court is still notable. If Democrats continue to get more unhappy with the Court, the potential to mobilise them against it will grow.
Much of the Court's public support is relatively "soft." We don't really know what would happen if a popular Democratic president (or, for that matter, a GOP one) made a determined effort to rally his or her party to the cause of court-packing or some other effort to neuter the Court. Nothing like that has happened since 1937. While FDR failed then, it is by no means clear a similar effort would fail today, given various changes in political conditions.
I should emphasise I am NOT claiming that the Court's popularity somehow proves the justices are doing a good job. As I have noted many times before, most voters know very little about the court's work (as is also true of much of the activity of the other branches of government) and public opinion is -at best - a weak indicator of quality. Similarly, the Court's continued popularity does not prove that Kavanaugh deserved to be confirmed or that confirmation process was handled well.