Fixing issues noted by ghc-pkg check
It's rather nice that ghc-pkg check will list broken packages, and why they are broken. But as far as I know, there is no automated way to take care of those broken packages. What is the recommended way to deal with broken packages? (Preferably not reinstall GHC)
Answer
Hopefully, you have been wise enough to not break any in your global package database. Breakage there can easily mean a reinstallation of GHC is necessary. So, let us assume that the breakage is restricted to the user package db (except possibly a package or two in the global shadowed by user packages). If only few packages are broken, you can fix your setup by unregistering the offending packages,
$ ghc-pkg unregister --user borken
that will often complain that unregistering borken will break other packages. Whether you try to unregister those first or unregister borken immediately with --force and deal with the newly broken afterwards is mostly a matter of choice. Make sure that you only unregister packages from the user db. If things aren't too grim, after unregistering a handful of packages, ghc-pkg check will report no more broken packages.
If, on the other hand, a large proportion of packages is broken, it will probably be easier to completely wipe the user db, $ rm -rf ~/.ghc/ghc-version/package.conf.d or the equivalent on other OSs.
Either way, you will have lost packages you still want to use, so you will try to reinstall them without breaking anything anew. Run
$ cabal install world --dry-run
that will try to produce a consistent install plan for all the packages you installed with cabal-install. If it fails to do so, it will print out the reasons, you may then be able to fix the issues by adding constraints to the packages listed in the world file (~/.cabal/world) - for example, although I have no broken packages (according to ghc/ghc-pkg), cabal install world --dry-run told me it could not configure vector-algorithms-0.5.2, which depends on vector >= 0.6 && < 0.8 (I have vector-0.7.1 installed). The reason is that hmatrix-0.12.0.1 requires vector >= 0.8. Replacing the -any "constraint" on hmatrix by a "< 0.12" in the world file produced a clean install-plan.
So, after a bit of fiddling with constraints in the world file, you will get an install plan from cabal. Check whether that would reinstall any packages you already have (installing a newer version is probably okay, reinstalling the same version means trouble). If you're happy with cabal's install-plan, cabal install world and brew a nice pot of tea while GHC is busy. Run ghc-pkg check once more, to verify all is in order.
A piece of generally good advice: If you don't know what installing a package entails, always use --dry-run first.
If you broke your global package database by doing global installs with cabal, the strategy of unregistering offenders may work, but it may also irrevocably break your ghc, that depends on what is broken in which way. If you broke your global db by installing packages from your OS distro, install a fresh GHC, curse the distro-packagers, and try to help them prevent further such events.
A cabal repair command would be very nice, but for the time being, repairing a broken setup is unfortunately much more work.