Circa 1800 - Age of Sail
The (fictional) <PERSON> made a voyage of roughly this magnitude of distance, with orders to stay out of sight of land until arrival, in a typical Royal Navy frigate (with a full crew), in the very early 19th century. The journey was from a West of England naval port to the western coast of Central America, via Cape Horn, and made a precisely accurate landfall. Though fictional, <PERSON> novels are well-regarded concerning historical plausibility.
In your scenario, I assume both continents have been sending out expeditions to each other and would have evidence of this, in the form of wreckage washed up on shore. This would constitute sufficient evidence (together with astronomical observations and the successful introduction of celestial navigation) that such a voyage was possible, even if nobody had yet succeeded, and probably also indicate the correct latitudes to pick up favourable trade winds.
NB: celestial navigation requires good knowledge of mathematics, an accurate observation instrument (eg. sextant), and an accurate timepiece (marine chronometer) if longitude is required as well as latitude. These were all available and in regular naval use by 1800.
Possibly even evidence of the length of voyage required could be obtained from the wrecks. If some of them failed only due to shortage of supplies, the ships might reach land intact but with nobody alive at the helm. Medical knowledge would be sufficient by then to approximately determine the time of death of the last survivors, and correlate this with the size of the crew and the quantity of supplies they set out with.
The ship's logs and charts would also be very valuable to future expeditions.
A civilian expedition would probably take a much smaller crew than a naval frigate, which would help to extend their supplies. 308
Not carrying a large gun armament and not requiring a manoeuvrability edge in combat substantially reduces crew requirements. Adopting a schooner (or topsail schooner) rig also helps, though it may reduce sailing speed in tradewinds. Since the winds and waves are likely to reach similar magnitudes as in the Roaring Forties on Earth, a schooner rig is almost certainly adequate.
Storing large quantities of preserved food on board is not a problem - salt pork and hardtack was the standard of the day, and would keep almost indefinitely. In roughly this period, pemmican was also introduced to expeditionary rations. Catching fish en route is a possibility, but probably should be regarded as relieving the monotony of diet rather than a primary source. Avoiding scurvy requires carrying a source of Vitamin C, probably derived from citrus fruits; concentrated lime juice became standard in the Royal Navy, and is not especially difficult to make.
The major problem is likely to be fresh drinking water, which is difficult to keep drinkable over a period of months, even if a sufficient quantity is taken on board. Depending on your planet's climate patterns, it may be possible to collect enough rainwater to sustain a small crew. The key technological advance that makes the expedition successful might be a relatively efficient solar-evaporative distiller, which could be made with a glass lid. Carrying spare lids would be advisable. 537