Well, hang on here. Are we talking about the percentage of the population that has a PC class? Or are we talking about the percentage of the population that makes a living by going into dungeons, killing monsters, and taking their stuff?
If the question is PC classes... well, let's start with the percentage of professional soldiers in the population. I haven't found any solid stats on this for medieval society, but figures I've seen range from 1-5%. Let's say 2%.
Now, looking in the Monster Manual, the Guard is probably a good example of the typical professional soldier. Guards are reasonably effective in numbers, but one guard is not at all a match for a 1st-level PC, even if you boost the soldier's ability scores to PC levels. So martial PCs are evidently the elite among the professional fighting class; maybe 1 in 5, which is to say, 0.4%.
But that's specifically PCs who are professional warriors: Barbarians, fighters, rangers, and paladins. (I'm not sure whether monks belong on this list or not, but to simplify things I'll say not.) That's 4 classes. There are 12 classes total, so if we triple that figure, we'd be looking at 1.2%.
If the question is professional dungeon crawlers... that figure will depend on how many dungeons the world has, but no matter how you slice it, it's got to be super low. If only 2% of the population is under arms to begin with, what fraction of that armed population can be spared to go poking around in monster-infested ruins looking for gold?