Purely saildriven ships were still in use until the early 20. century. Most of them were used for relatively low value bulkgoods (often but not always!) because sailing ships still had some advantages over coal driven ships, which required expensive machinery and a reliable supply of coal. The large teaclippers and later the Windjammers, made possible by introducing iron/steelhulls, are excelent and widespread examples for mid/late 19th century sailships which, at their time, were economicaly viable. Once steam engines became more refined the large sailships slowly fell out of use but by then we are well into the 20th century.

I currently don't have any exhaustive literature about late 19th century ocean going sailing vessels (perhaps the last chapters of Chapelles The Search for Speed Under Sail, 1700-1855 would qualify) but on a small scale the same development can be seen in the area of inland navigation. Here Fontenoy's The Sloops of the Hudson River provides an excelent example for the challenges but also the economic niches that came with the use of sail driven ships/boats.

Hi there, I've written a few answers on this in the past that may be of interest to you:

https://www.reddit.com/r/AskHistorians/comments/2gv83n/what_sort_of_jobs_did_the_crew_do_on_an/

https://www.reddit.com/r/AskHistorians/comments/2jso46/how_did_the_transition_from_wooden_ships_to/

https://www.reddit.com/r/AskHistorians/comments/28yrxj/did_the_unions_monitor_have_a_unique_design_for_a/

To quote a bit from those answers:

The first oceangoing ironclad vessel was the French La Gloire, whose keel was laid in 1858. It was essentially a frigate with iron plates bolted to its timbers above the waterline. The British navy responded with HMS Warrior, launched in 1860. Warrior was an oceangoing ironclad that was both full-rigged for sail and equipped with a steam engine, which made her the precursor of many hybrid sail/steam ironclads.

Some combination of La Gloire and Warrior were the likely inspiration for the Confederate conversion of USS Merrimack into CSS Virginia, which had one day of glory when it sank USS Cumberland, burned USS Congress, and drove two other ships aground. The Virginia's design was substantially different from earlier ironclads in that the hull above the waterline was reconstructed and it depended entirely on steam power.

The Monitor was created as a design specifically meant to counter Merrimack/Virginia, and it was unique in having a revolving turret, which meant that it did not have to maneuver to bring its guns to bear. It was not, however, meant to be an oceangoing ship (nor was Virginia) and in fact it sank under tow off Cape Hatteras.

and from another answer:

The 1880s were an interesting time in ship design, because the traditional divisions of sailors were changing to accommodate the advent of steam propulsion. Most of my comments will apply to the British and American navies, since it's what I'm most interested in. Navies had since time immemorial been powered by sail, and many early steam vessels also had masts and a full sail rig . This should not be confused by the mast or masts on a modern ship , which are used mostly for signaling or as sensor platforms. Ironclad masts were used as a primary or secondary power source, as well as a place for sail drill. This was in many ways a logical design evolution, when capital ships were converted to add steam power, though it looks odd to modern eyes.

In any case, the addition of engines meant the addition of engineers, which meant a change in the traditional rank structure of navies and their internal organization. Initially, officers resisted the addition of engines and dirty coal to their ships, even as it was widely recognized that coal and steam power were the future of navies.

(Apologies for the enormous image link , but Turner's "The Fighting Temeraire Tugged to her Last Berth to be Broken Up" is highly symbolic of this understanding.)

Anyhow, the addition of engineers to the traditional structure was a source of tension in many naives. Conservative elements in the British and American navies assigned engineers initially to the ranks of warrant officers, and only reluctantly added them to the ranks of actual commissioned officers. Conversely, up-and-coming officers of a technical bent saw engineering as a growth field, which led to a great deal of tension among younger and older officers.

In terms of internal organization, by the end of the Napoleonic wars the British and American navies had adopted a divisional system, in which an officer was assigned to a set of men on the ship, so that even in a very large ship each man was assured of being known to his own officer. The men were further divided by stations on "watch" by their skills, with the most prestigious jobs for a seaman being work aloft (although there were many other specialized jobs having to do with all stages of running a ship). The change in propulsion attacked directly the boatswain's department, as being in charge of the sails and rigging, and also the self-identity of the Navy. Officers wondered how men would be trained without constant work on the masts and sails, and reactionary elements in the British navy continued to present papers and arguments in favor of sails though the 1890s (sails were officially abolished from new construction in 1887).

In terms of the actual jobs done, sailors tended to do many traditional jobs: scrub decks and fittings, exercise with the guns, hoist in and out boats, re-stock the ship with provisions, take care of compartments, tackle, sails, flags, etc.; but the new jobs done were coaling, maintaining the ship's machinery, and understanding the complicated mechanical changes brought about by boilers and engines and steam power. The men would be organized into divisions under a particular officer, with warrant officers supervising individual work spaces and/or groups of men, who stood regular watches.

Some sources on the transition that are interesting reads:

Sacred Vessels: The Cult of the Battleship and the Rise of the U.S. Navy

The Rules of the Game: Jutland and British Naval Command

I think the question really needs to clarify whether you're asking about military or civilian use of sail. The Star of India was by no means a "late" sailing ship in 1863. Both the Balclutha (1886) and the C.A. Thayer (1895) are sailing ships that enjoyed long careers hauling cargo and passengers.

These ships were still very economical well into the 20th century largely because the labor required to run them was quite small compared to warships. The Balclutha sailed on its maiden voyage with a crew of 26 compared to the hundreds that would crew on a warship (to man the guns, fight the ship, etc.), so a voyage that took longer via sail would still be more profitable than the fuel to power a faster boat. The primary cargo carried was (originally) California grain to the UK and coal on the return. Because these goods were relatively non-perishable, there wasn't the emphasis on speed versus the overall profitability of each voyage regardless of the length. (Sidenote: sail on the Pacific coast lasted longer than in other areas because there was no good source of coal; the early development of oil drilling in California also led to earlier adoption of oil powered steamships as well).

The C.A. Thayer was primarily a coastal vessel, sailing up and down the Pacific Coast of the US hauling lumber. Again, non-perishable cargo where speed mattered less than cost of operation.

Later in their careers, both ships operated well into the 1920s supporting the salmon fishing and canning trade in Alaska, hauling labor and fishermen up and canned salmon down.

Sailing ships were also in use well into the 20th century on inland waterways. The Alma plied the inland waterway from Sacramento area to the main ports of the San Francisco Bay hauling cargo under sail until it was converted to a pulled barge once oil became more common in the 1910s.

There's no question that steam overcame sail much earlier in military ships, both for the consistency of travel and maneuvering regardless of the wind and for the labor savings of having more men available to man the guns instead of shifting sails in battle (plus the fact that the purpose of those ships was to sink enemy ships and not get sunk, a very different economic model!). But for commercial vessels, the economics of sail lasted far longer.

By the way, if you're ever visiting San Francisco, all three of those ships are on display (along with the 1907 oil powered, ocean-going steam tug Hercules and several other vessels) at the San Francisco Maritime National Historic Park.