I saw a video of a philosopher (Robert Audi) who said that common
sense is the best response we can give to global skepticism. I would
agree, but it's not clear to me what the nature of common sense is. Is
it a set of intuitive beliefs?
No, it is very rational. But in order to see it, we need to take a step back out of the epistemological haze; and maybe even to step back further out of philosophy... as it exists now, anyway.
We need to take a look at the big picture and ask ourselves, "What is the point?.." Why do we need philosophy in the first place? To know the nature of truth? -- but why? Why do we need to know it? Or the nature of reality?
More than two millennia ago Socrates, apparently, found himself in similar circumstances when he felt the needed to remind his interlocutor:
"Do not take what I say as if I were merely playing, for you see the subject of our discussion -- and on what subject should even a man of slight intelligence be more serious? -- namely, what kind of life should one live?.." -- Plato, "Gorgias"
In other words, the ultimate question, therefore, is never about some abstract knowledge, like the nature of truth. Rather, it is about its practical implications -- the latter being more important than the former. It means that, sometimes, the most rigorous model and, technically, the most accurate answer will be rejected simply because its inferior practical implications.
The above concept is also known as the Streetlight Effect, so named after a parable of an emphatically rational drunkard, who, after losing his keys the park, was found searching for them under street lights.
Here is a video of a crow, apparently relying on the "streetlight effect" to start solving the puzzle before he knew the full solution.
Back to global skepticism
I'll focus the Cartesian model, specifically, because it is the most rigorous theory of truth, reality, and knowledge. According to it, we cannot know anything beyond the only knowledge absolutely must have -- that of our own existence. And, strictly speaking, this is it.
Unfortunately, we cannot accept it, even tho it is perfectly accurate, because of its practical implications. Having no knowledge of the outside world leaves us with no freedom to do anything. There would nothing to do until we die from thirst or sooner.
That leaves us with Søren Kierkegaard's leap of faith as the only way out. We have to come up with assumption(s) that would let us develop a more useful theory of truth.
As it turns out, one assumption is all it takes: we assume the existence of one and only reality, explainable through lógos, or logic and reason, which we all share and are a part of (a.k.a. the one and only God of pre-civilization, fully rational humanity).
The above makes for the first premise, which a fully rational system of beliefs can be inferred from trough logic and reason.