Consider how we may theoretically obtain information:
- Directly, through our senses.
- Built into our design, EG: instinct, reflexes, hormonally/pheromonally triggered behaviors, etc.
- By rational analysis, principally induction and deduction.
Encoded, transmitted through our senses and then decoded. Language and most everything we've "learned" would fit this category.
We distinguish this from normal/direct sensory data by noting that our bodies do not naturally know how to decode this information; we have to be taught how to consume it.
For completeness, we can't disprove that divine revelation/inspiration is possible.
Given all that, it should be obvious that "empiricism versus rationalism" is a false dichotomy. We get knowledge both ways and more.
As for, "What extent, does our knowledge derive from our senses?" Whole books can and have been written on just aspects of this, the question is very broad.
Consider a typical scene in our moment-to-moment existence, we might learn the following "facts" (probably automatically and without consciously realizing/verbalizing them):
(a) The remote is there, while (b) the beer is there, but (c) the wife is standing there. (d) She's holding a garbage bag. (e) I feel tension in my jaw.
In that scene, we know quite a bit directly from our senses, but most of our sensory data is filtered into responses and/or concepts automatically, then the raw sense data is mostly discarded. The individual photons and pressure waves are vital, but do not matter in and of themselves.
Note that we all can (and do) trust our senses with our very lives. Sure they have limitations and can be tricked, but these limits are almost always outside of what is required for our day-to-day survival. We would have long ago been crocodile chow were it otherwise. The limits of our senses are now known; they are predictable, and consistent across individuals and species.
Now back to that scene. The individual photons, sound waves, etc. were sensed, but it was how we accurately interpreted them that mattered. All animals have innate knowledge built-in to help interpret their senses into the fundamental (to life) concepts of the need/opportunity to: fight, flee, feed, or f..., er, fornicate.
It's automatic, but it is knowledge, and it is objectively right most of the time (or our parents wouldn't have lived long enough to breed). This knowledge was designed into our bodies by evolution -- a kind of meta induction.
So, at a fundamental biological level, the most important knowledge we have is not sense-percepts, but concepts that Mother-Nature inducted for us after collectively processing lifetimes of sense-percepts.
Even the existence and design of our senses is a kind of built-in knowledge. One thing that biology and neuroscience is teaching us, is that there is not "mind" versus "body", and there is no "inner control room" to the mind. A mind cannot exist without a body, and the whole body affects the mind (and vice versa). "Thought" and "knowledge" occur across a spectrum as a process; the structure of our brains greatly affects what we "know".
Studies of siblings suggest that "Nature" is still a substantial part of who we are and that "nurture" can effect only so much.
So, as animals, much of our knowledge is built in. We learn things via our senses, but it is how we interpret our senses, via deduction (most of it automatic) that keeps us alive.
As humans, we transfer massive amounts of information to each other via training and language. Initially, this information would have been concepts inducted from sensory observations by our ancestors (Or prepackaged by God, if you wish).
But, then new deductions could be made from those concepts, initially devoid of sensory data (discoveries in astronomy and physics have been made this way). However, in Science, such deductions are worthless unless they are eventually supported by sensory data.
Sensory perceptions are the bedrock upon which all of our (useful, verifiable) knowledge is built. But, the bulk of what we currently know and use was long ago inducted into concepts or into the very design of our bodies and minds. Today, concepts, innate and otherwise are the dominant factor in our knowledge and in our survival.