Fresh water was a big problem, and so was scurvy.
On a 15th century sailing ship, pretty much they didn’t distill any water. Even if they had the technology, they couldn’t carry enough fuel.
Going back to the OP I think the problem is defining “the average Joe”.
As others have already alluded to there was a much sharper divide between the “educated” and the “uneducated” and the proportion of those with a decent eductation would have varied from place to place. I don’t know about Genoa or Castile but in England schools were being founded in the 15th century - available free or at low cost to all local boys - but the big increase in grammer schools wasn’t until the 16th century so the “educated” man would have been pretty rare in the years before Columbus sailed.
FWIW, some average Joes today are not convinced that the world is round.
YouTube video with sound.
As others have noted, they meant both. It was a considerable challenge to keep a crew healthy on a long voyage. The necessary knowledge wasn’t gained until the late 18th century - and Capt. Cook’s empirical understanding of how to prevent scurvy was slow to catch on.
There was no practical distallation of seawater on sailing ships - this had to wait for steam power.
If the question is asked, instead, “How sharply curved is the Earth”, then it’s pretty difficult to answer that it has a nonzero curvature, based on the kinds of local observations I can think of doing.
That bit about ships visually disappearing downward as they sail away - when is that true? Whenever I see ships on the ocean, as they get further away they become indistinct. It’s not always apparent which of several horizontal lines and gradations is the horizon. Of course, the effect should happen geometrically, but as a real visual effect how many people here can say they’ve witnessed it?
I’ve seen it thousands of times - it’s essentially always apparent when the visibility is decent. Sailors routinely speak of a ship being “hull down” - you can see the upper part, but not the hull.
I’ve seen it many times, as I am sure have many other people who have spent any time on the sea. It is not just ships, of course - when you are approaching land from the sea, if there are any hills or mountains you will see the peaks first. This is quite obvious. I’m sure that in Columbus’s time mariners at least would know that the Earth was curved, if not a globe.
Here is a picture of a globe, from right around 1492: http://www.henry-davis.com/MAPS/LMwebpages/258.html
The accompanying monograph notes:
Well, there’s the fact that at a given latitude you get essentially the same view of the stars no matter what your longitude - the only difference being the time at which a certain view is available. Unless you postulate that those stars are really close (which has its own problems), it’s hard to see how this could be true with zero curvature.
You may be thinking of modern ships where the upper part of the superstructure is not substantially different in shape from the lower part, so it’s hard to tell how much of the ship you’re actually seeing. The effect would be more noticeable in a sailing ship with a tall mast or masts, which would remain visible after the hull disappears.
From what I’ve read, you should cut Columbus some slack. Agreed, pretty much everyone involved thought the world was round. The natural philosophers, or whatever they were called at the time, knew the dimensions pretty well.
But, Columbus was one of the greatest sailors ever. He based his estimate for the size of the earth on debris washed up along the coasts of Europe and Africa. His reasoning was impeccable. “Given this much unknown biological stuff, land isn’t so far an expedition can’t get there.” IIRC, his estimate for the distance he had to sail wasn’t too far off. He sold Isabelle on taking a chance on the distance, because of his evidence and reasoning. (He was also a great dead reckoner, but that is an aside.)
Of course, damn if I can remember where I read this.
IIRC Portugese explorers heading south along the coast Africa hit the point where the (northern) pole star disappeared below the horizon and the lack of that point hindered exploration. At any rate the non-visibility of Polaris and the northernmost stars in 1471 when the Portugese explorers crossed the equator would have most likely clued them in any case.
His estimate of the distance to be traveled was way way off. He thought the distance from western Europe to Asia heading west was less than 3,000 miles, when it is actually about 11,000 miles. That’s a pretty damn big error. It was only the fact that the Americas were where he calculated Asia to be that his expedition didn’t end up an utter disaster or coming back without result.
Thanks for the link, Gfactor.
No problem. I’m currently working on a Columbus-related staff report. I had to look it up for that anyway. 
I thought that the point of SlowMindThinking’s post was that Columbus was betting that there was a landmass not so far away. Columbus’ mistake was thinking that the landmass was Asia when in fact it was the Americas. So he was wrong on the continent but lucky in his deduction, from the debris washed on European and African shores, that another landmass was not so far away. Now, if his deduction had a rational basis, that I don’t know.
(This is of course assuming that what SlowMindThinking is saying has a historical basis.)
Link doesn’t work. I keep on trying to rotate the globe with my mouse pointer but it won’t budge. 
Thanks Colibri. Of course it’s obvious when you point it out. One last question though - the Polynesians were travelling a third of the distance Columbus did, but were they travelling as fast? I imagine that Columbus’ ships would travel faster than a Polynesian ship from 1000 AD. What were the Polynesian ships? Catamaran-type ships (I seem to recall reading that somewhere) or just big canoes?
This page (evgschool.org) says that Columbus’ ships covered approximately 150 miles a day. I couldn’t easily find an estimate for the speed of the Polynesian vessels.
Rotate your monitor.
It would depend a lot on conditions, but on average they were probably travelling somewhat faster. The large ocean-going outriggers had much smaller sails than European ships but much a much shallower draw, so under conditions of light winds they would have moved faster. As wind speed increased the European ships would have gained the advantage. Outriggers also have a massive advantage sailing close to the wind, so if the voyage is running against prevailing winds the outrigger wil travel much faster.
Cook wrote of his one of his contacts with Polynesians that their large outriggers were quite literally sailing rings around him by alternating between just sails with the wind and supplmenting with oars against it.
Both, the vessels for long voyages were almost always outrigger canoes. They weren’t strictly catamarans since they had only a single hull, but one or two logs were attached to that hull via poles to produce a very stable outrigger. Cloth was then stretched between the hull and the outriggers to provide extra storage space.
By using only a single hull to minimise contact with the water outriggers are much faster than catamarans of the same size.
"Voyaging canoes in the Caroline Islands have reportedly made long passages at average speeds of as much as eleven knots and larger Marshallese canoes have been clocked at 16 knots. "
So at 11 knots that’s about 303 miles/day if that speed can be maintained constantly, and about 150 miles/day if only sailing for 12 hours/day. The actual figure is presumably somewhere in between if we assume a high speed is maintained during daylight hours and a more sedate pace at night. Certainly comparable to Columbus’ speeds.