As in, the crew didn’t have to worry about getting scurvy or running out of food or getting lost at sea etc…?
I guess “safe” in this context subjective but hopefully you get the jist.
I’d say when steam travel became common. Sure, you could still have boiler accidents, but not being left to the whim of the winds was a big turning point. Travel from Europe to the Americas may have been fairly safe before steam, but world wide the steam engine was a game changer.
Another helpful factor was the introduction of the first reliable chronometers beginning in the early 19th century, which made navigations less of a crapshoot than it had been previously.
Scurvy - Wikipedia is an interesting subject. Even after the realization in the latter 18th century that something about fresh food seemed to prevent or cure scurvy, soldiers, sailors and explorers continued to suffer from it even into the beginning of the 20th century. It was only in 1907 that the first animal model for scurvy was discovered, the aptly-named guinea pig. What we now call Vitamin C wasn’t finally proven to be the active ingredient against scurvy until 1932
I have no empirical evidence to support this, but…
I would have said that it became pretty safe around the Napoleonic era, especially for British ships. It was relatively common by then to entrust family members to ships to reach the many and varied colonies. The idea of sending, for example, three ships on a long voyage and losing at least one of them was pretty much behind everyone at that point. When a ship was lost at that time, it was pretty safe to assume that it was enemy action, which could also be said for the WWII era.
Ben Franklin made two lengthy trips to England prior to the Revolution and then went to France as US ambassador after the war started. These weren’t considered esp. hazardous or unusual trips at the time. Other US notables also went to Europe for diplomatic or business reasons.
I think one of the key things was the reliability of the ships. The tech had evolved by that time that ships could routinely make round trips across the N. Atlantic over many years. Not at all like the poor ships Columbus or the Mayflower people used.
OTOH, that wasn’t the same as crossing the Pacific. But even there they had regular galleon fleets that sailed yearly between the Philippines and Mexico. Enemy action caused more problems than storms.
Losses due to storms continued to be a problem for a long time. Shoot, a typhoon during WWII did a lot of damage to Halsey’s fleet.
Notice that the chemical name for vitamin C, ascorbic acid, means anti-scurvy.
I would expect it also mattered where you wanted to go: for some areas you might have reliable, consistent trade winds–and for other areas you didn’t.
When the Titanic was launched!
EDIT: D’oh!
It takes about three months to get scurvy. It took the Mayflower less than 70 days to reach the Americas in 1620. By the early 19th century it was taking less than a month for some of the slower sailing ships to cross the Atlantic. I think probably by the 17th and 18th centuries you were pretty safe on a sailing vessel so long as you weren’t exploring. Stick to established trade routes and you were probably fine.
I would suggest not until after WWII when submarines tended to be a fairly large hazard.
Just my 2¢ worth. 
Lucy
It would depend on the starting point, wouldn’t it? Likely a large portion of historic seafarers didn’t have healthy levels of vitamin C intake before embarking on an ocean journey. They could get scurvy much faster.
It took a while to develop reliable steam engines. The early ocean faring steamships still had sails just in case. The combination of a steam and sail power was still better than being solely at the mercy of the wind.
I thought the big thing was whether or not you were staying within one of the prevailing wind bands and not going through the calm areas at roughly 60 degrees N/S, 30 degrees N/S (the “horse latitudes”) or at the Equator (the “Doldrums”).
Between these areas are the prevailing wind bands- from the equator to 30 N/S there are the “Trade winds” which are prevailing easterlies (i.e. blow from east to west), and from 30-60 N/S there are the prevailing Westerlies, which as you might have figured out, blow from the west.
Sometimes ships that were a bit unlucky got stuck in the horse latitudes or doldrums and were becalmed, since those latitudes aren’t as windy as the trade winds or westerlies.
Interestingly enough, we can see this in action with Atlantic hurricanes. 30 North is roughly where the Gulf Coast is- Houston and New Orleans are both at that latitude. So when we see hurricanes heading westward, that’s the trade winds blowing them east. Coriolis forces direct it northward, and then as it crosses 30 N, they start curving eastward, due to the prevailing westerlies.
As far as sea travel goes, if you’re in the tradewinds or westerlies, you generally are going to have wind, and can go either direction. It’s going between them that got people in trouble.
One thing that hasn’t really been mentioned yet is the advent of storm tracking and weather forecasting. A big enough storm will sink even a large, modern ship. This was seen less than ten years ago when the large cargo ship El Faro sank with all hands when it sailed into the eye wall of a hurricane in 2015.
@ftg did mention the typhoon in 1944 that sank three U.S. Navy destroyers and killed 790 sailors. This tragedy resulted in the establishment of the Joint Typhoon Warning Center.
The reason this happens far less frequently these days is because ships take steps to avoid storms. When they don’t, bad things can happen, as the captain and crew of the El Faro found out the hard way.
Wouldn’t one say e.g. Cape Horn was stormy? That’s the impression I got from Herman Melville, Henry Dana…
After the Titanic disaster, safety reforms were instituted to make sea voyages safer, including having enough lifeboats for all passengers.
“Relative” safety in terms of ships avoiding obstacles including each other was markedly enhanced by having radar on passenger vessels starting in 1942, though that didn’t protect them from submarines or ineptitude in interpreting radar (as happened when the Andrea Doria sank after a collision with the Stockholm).
The Phoenicians and the Romans were transiting the Mediterranean in classical times, if that meets the definition of ‘sailing across the sea’ and it was done on a commercial basis, as Egypt was Rome’s granary as well as supplying papyrus, decorative stone and minerals. There were risks, of course, but it wasn’t pioneering stuff.
As mentioned earlier, by the 19th c. two thousand years of cumulative experience in ship design, and the beginnings of an understanding of stability, in terms of ‘the power to carry sail’, had developed. The metacentric theory provided a useful approximation of what was a good design, and what was not, although until computers began to become available in the 1950s to do the number-crunching, really detailed assessment of stability was so lengthy as to be deemed impracticable.
Until the compound expansion steam engine reduced coal consumption to the point where really long (trans-Pacific) sea voyages were possible, sail still had an advantage.
Sure. But we’re defining relatively safe as per the OP where one could be expected to sail without suffering from scurvy or getting lost. There’s always some element of risk. Even modern cruise ships have some elements of risk including sinking and passengers catching some God-awful disease in their floating petri dishes. And depending on how one is defining sailing, the question of safety often boils down to where you’re sailing. The Mediterranean is relatively calm and various peoples didn’t seem to have a problem sailing it in the ancient world. Scurvy wasn’t a problem because they generally didn’t stay out at sea that long and they typically hugged the coast unless they had to cross and weren’t likely to get lost.
Sailing ships being unable to move directly into the wind, the dreaded “lee shore” that a ship in stormy weather could be driven upon and smashed was a concern.
Regarding scurvy: it’s estimated that some 2 million sailors died between 1600 and 1800 from the disease. An early proponent of a varied diet at sea was Captain James Cook, who apparently never lost a man to the disease. While neither he nor anyone else knew the causes of scurvy, Cook realized that his men were healthier when they ate whatever vegetables and fruits were available on his voyages in 1768 and 1769, in addition to their normal seaman’s diet.