The Closest 'Real World' Equivalent to DnD Adventurers is the Noble Class
One of the best, and only, ways to enter the Noble class from the peasantry was to be so ridiculously puissant at combat that you were given a command - and from that tiny band of men, achieved so much so prominently and notably that you were given a higher rank, and so forth. Saving the Duke personally on the field of battle, etc, was a quicker version of this.
In DnD, where individuals have the firepower of an entire battalion of cavalry, Adventuring is the fast-tracked version of that. The ambitious, the talented, the driven - those are who adventure. 'Adventure' is really a misnomer - those are who grasp the rungs of power with both hands and delve deep beneath the earth into a god-king's tomb full of the corpses of those who have tried on the 1% chance that they'll learn a new spell or find a +1 sword.
That said, in any realistic setting (medieval or otherwise), everything is geared towards competitive advantage - in the case of settings of mostly humans, this means Nations. Although the idea of a Nation is less deep-set in a medieval era than one of our modern educated nuclear-equipped ones, ultimately there has to be some sort of structure to keep out the Mongols - and so any magical power (+1 swords) or powerful individuals (adventurers) would need in some way to be tied to the State.
And so we come back to Nobles. Where in the medieval era they were trained to be officers and heavy cavalry (good officers and heavy cavalry ruled the battlefield in those eras) and paid in land, in DnD they can literally cut a dragon in half with a sword, so they're probably paid in 'whatever they want' and are vastly more important than even the nobility were to medieval nations.
Wandering Bands of Armed Men existed in medieval europe - they were called Bandits
Or mercenaries, and there's a pretty thin line between the two. Some people leave home to become soldiers and desert, or the crops fail, or whatever, and they end up bandits. There were a lot of them all the time, they were desperate, and tended to get murdered whenever anyone's armsmen or bailiffs caught up with them.
Likely though, men broken by circumstance wouldn't be the ones fighting terrifying monsters. But Adventurers could be viewed in a similar light - even, both, with low level or unknown adventurers being seen as little better than vagabonds or mercenaries, and high level ones being treated like rockstars and courted by noblemen's daughters.
Ultimately, people are greedy and scared
People will do whatever benefits them, or what they think will benefit them. This is rarely very logical. People bow to power, of any kind, if they can identify it. People try to better their station, even if it is already exalted. People will demand protection and safety, or beg for it, whichever they think will work.
There's a lot of roles for 'adventurers', from Monster Hunters to Landed Nobility to Mercenary Scum. It could, and should, differ from culture to culture (think about it - Romantic France Adventurers vs Byzantium Adventurers - how would they be treated?), but regardless of the differences, there should always be this - they should always be Important. People respect and fear power, and someone who can cut through a steel door with a teaspoon? He's powerful, and he's scary.
How a nation treats it's powerful individuals should be a clear and defining feature of that nation, even if it pretends to scorn them rather than giving them lands and handsome men.
Sources: Setting building, Frank and K's Races of War, common sense