Modern passport and visa controls were introduced during the First World War in Europe. I cannot speak for other regions. Although initially it was an effort at keeping enemy subversives out, after the war, European governments became concerned with who was entitled to the social services that were becoming increasingly popular among European governments. This necessitated a way to differentiate between citizens and foreigners, something that had not previously been of great import to European governments. Instead of worrying about "foreigners", most European governments were more concerned with drifters, vagabonds, and other rootless and poor people, regardless of where they came from.

Keep in mind, that until the advent of mass travel via the locomotive, immigration control would not have been a huge issue for any European government, as there simply were not enough people making international trips, and those that did, tended to be wealthy - think aristocrats on the grand tour. From the initial introduction of locomotive travel until the First World War, European governments did not concern themselves with such matters, primarily because they were still worried about the rootless poor, but also because there was not the necessary state infrastructure to adequately handle these sorts of issues (I won't use the word "problem" because I don't think it would have been viewed as such).

As for passport-like documents - i.e. documents requesting safe travel by the issuer for the bearer in foreign lands - those date back, at least, to biblical times and would have been issued by various governments throughout history to individuals deemed worthy or those for whom it was deemed necessary. Nehemiah 2:7-9:

I also said to him, “If it pleases the king, may I have letters to the governors of Trans-Euphrates, so that they will provide me safe-conduct until I arrive in Judah? And may I have a letter to Asaph, keeper of the royal park, so he will give me timber to make beams for the gates of the citadel by the temple and for the city wall and for the residence I will occupy?” And because the gracious hand of my God was on me, the king granted my requests. So I went to the governors of Trans-Euphrates and gave them the king’s letters. The king had also sent army officers and cavalry with me.

The term "passport", according to this Guardian article, dates to the early modern period, when the British government began issuing documents to help travelers obtain safe conduct while overseas. Only in the 19th century did their usage as identity proving documents begin, and again, only with the First World War did they become necessary to cross borders.

Sources: Cliff Rosenberg, Policing Paris: the origins of modern immigration control between the wars covers my claims from the first two paragraphs. The third paragraph is a direct quote from the bible (New International Version), and the last is based on that Guardian article that's linked, so yeah. Don't go asking me for more sources, cause I ain't got 'em.