The white walkers are a constantly-looming threat over Westeros, but in a worst-case scenario where the wall is breached and Westeros falls to the white walkers, would Essos be similarly doomed? Would the Narrow Sea protect Essos from the white walkers, or do they know how to build / pilot ships?
Yes Essos will be threatened as well.
Long Night occurred some 8,000 years ago but it was not limited to Westeros only. Legends point to the same event all over the Planetos as there are similar legends found as far as Yi-Ti and Asshai. Melisandre knows about this event even though she is from Eastern most land of Asshai. If the others managed to threaten the Eastern realms back then, it is likely that they would be able to do so again.
According to GRRM in an interview:
“The Others can do things with ice that we can't imagine and make substances of it."
In S06E10 we saw that Citadel sent White Ravens to all lords, which heralds the coming of winter. Winter means more ice and snow and thus increase in power of the Others.
Using their power of creating substances out of ice, they might be able to build an ice bridge across the Narrow sea (It is called Narrow for a reason) and wage war in East and West.
We do not know exactly how can they threaten Essos but we do know that they did so in the Past and thus have the power to do it again. Remember, Arm of Dorne, the sole land link between Essos and Westeros was broken some 4,000 years before the Long Night fell. Others managed to extend the war beyond the sea anyways back then and there is no reason why they should not be able to do it again.
Since the whole world has not been explored, and it is not known what is beyond the Land of Always Winter in the north, or beyond the Grey Waste in the east, there is no way to know if these two continents are in fact connected as one. There is the possibility that they connect beyond the edges of the map.
It is a huge world. If Sothoryos is reportedly at least as tall as Essos is known to be wide, then it's just as possible Essos and Westeros are actually one continent. The Others could be the true threat that the Five Forts were built to protect against, and if the Others came from that direction, would suggest such a continental connection.
This might explain how the Others threatened Essos in the past. The short answer is that they did threaten Essos in the past, so likely would again this time around.