Short answer: It wouldn't change the Constitution at all.
Longer answer: If it had been known when Arthur was in office, there would have been legislation and contemporaneous commentaries, which wouldn't have changed the Constitution, but would have shed some light on in how it was interpreted at the time. There would also have been litigation, which would have set some sort of precedent.
If one unqualified president slipped through undetected, it doesn't change the law, any more than one killer who isn't caught invalidates the murder statute.
A precedent that would have been in place for today's Supreme Court to refer to.
So, do we owe it to future presidents, courts, and voters to set that precendent once and for all, so it doesn't happen a second (or possibly third) time?
-PJ