Are there any classic proofs of the necessity of a mind-independent "reality," along the lines of Anselm's proof of God?
There are no popular theories pertaining to the "necessity" of an external world. Bishop Berkeley tries to prove the necessity of an entirely mind-dependant world, but direct realists have only refuted their opposition, and not made any solid attempt to defend reality. The only way to have necessary existence of reality is to invoke God as Descartes does when asserting his own existence, and invoking the non-natural opens a theory to more problems than it solves.
You might be interested in Paul Boghossian's Fear of Knowledge. As I recall, though, I did not find his argument for an objectively existing reality (I will use "OER") convincing.
I will point out that the moment one talks about other people (or, for that matter, tables or chairs) he is assuming that there is an OER. If he wants to avoid that assumption, then he has to talk about his own other-people-ish sensory qualia (or about his own table-ish sensory qualia or his own chair-ish sensory qualia). I will also point out that to say that science has the virtue of its hypotheses' being testable once again presupposes OER. Thinking of yourself as interacting with other people (or with tables or chairs) presupposes that you accept that there is an OER (as well as that you derive reasonably reliable information about it, presumably via physical senses). Unless you are willing to restrict yourself to talking about only your own sensory qualia, it is a practical necessity to assume that there is an objectively existing reality of which you gain reasonably reliable information via some informational connection (presumably, physical senses). And to see other people as having moral value, or thinking that they ought to be so treated despite morality's not being objective, requires thinking of them as existing beyond your own mentality--and therefore requires assuming that there is an OER of which you somehow gain reasonably reliable information.