For a part of Korea, both France and the US sent punitive expeditions after the Joseon killed a fair number of people from both countries, though they didn't lead to much, due to insufficient forces and lack of political will to impose anything too big. If either affair escalated a bit more, cessation of a treaty port probably would've been possible and that'd fulfill colonizing part of Korea, yes?
As for full annexation, that's a bit trickier due to Korea being important strategically for both Asian powers, the Qing and Japanese Empires (a knife pointed at the heart of Japan, after all). Both the Qing and Japan have to do significantly worse than OTL, that's a given, for Korea to be annexed in the 19th century.
Wikipedia says that one of the French commanders during the French invasion wanted to force the Joseon king to cede sovereignty to the French but there's no citation on that so I'm not going to count that. The main European that might colonize Korea would be Russia but they'd have to chew through Manchuria first to make it worthwhile logistically.
Let's say Qing China does worse in the Taiping Rebellion, the Tokugawa hold a bit better by playing the tozama daimyo off of each other (Satsuma and Choshu, for example) and the Boshin War stretches out longer, enough to impact modernization and industrialization overall. Russian influence in Korea grows paramount over the decades, leading to a faction of the court turning to the Japanese (as the Qing have been shown to be unable to protect themselves, let alone Korea) to counter Russian influence. Tensions escalate and eventually the two go to war over dominance in NE Asia. Due to weaker infrastructure and poorer economic backing, the Japanese concede and the Russians declare suzerainty over Manchuria and Korea, formerly annexing them. Can't see it lasting, though, since administration over a highly populated region on the other side of the continent requires either a good navy or the Trans-Siberian Railroad and will in both cases is horribly expensive, uprisings would be difficult to put down when the Polish are also militant, and Japan would have to be put down as a threat permanently or an alliance by the British, Germans, and Japanese would break the Russians due to how divided their forces would have to be.
There is functionally little difference between the Japanese colonization of Korea and the Belgian colonization of the Congo. Japan, like Thailand, out Europe'd Europe, and they were incredibly similar in their ambitions and goals... and approaches. Or in plainer terms: They were colonized by a Western nation.
Western is not geographically based, it's ideologically based.
Eh, the Belgian Congo's not that apt a comparison. French Vietnam, perhaps, but the Japanese weren't cutting off our hands for not making quota. Not that the Japanese weren't brutal in their own ways, especially later, but there's quite a few nuances to be found. Like with the matter of ethnic and cultural affairs, the Japanese tried to impose the 'same ancestor' root (they said they were the better branch, big surprise there) and assimilate the Koreans in a way that Europeans could not do in their overseas colonies (think Russia with the other Slavs, basically). They were much more hands-on than most colonial overlords, though that's a matter of proximity (more like French Algeria than French Vietnam) but that's a big difference between colonization by a European empire vs Japan for Korea (the former would have to be much more hands-off just due to bureaucratic costs) and that'd cascade into culture (the Japanese interpreting and rewriting parts of Korean history to fit a perpetual Japanese domination narrative, racial purity and superiority a la North Korean Juche would not be quite as dominant in culture without the Japanese use of it help assimilation), economics (industrialization under Japanese rule was quite different from the typical colonial economic policy under European overseas overlords and did define quite a bit of Korean economic history), and diplomatic relations (not having quite as much animosity towards Japan and not having anti-Chinese propaganda imposed would drastically alter Korea's relations with the rest of NE Asia).
As to any Western nation being drawn to Korea. Outside of Russia - and they would have to make a move no later than 1880's - I can't imagine much interest in a country that doesn't have significant exploitative resources for the capital, military as well as economic. that would be needed.
Korea, politically, economically, and socially, is a Japan/China story, not much of a Western one (still is, the US is just being Japan's surrogate in the North Pacific)
For the former point, Korea held a fair chunk of iron, coal, gold, and a strategic point in Asia that would help dominate trade and sealanes in NE Asia. Parts of Africa were colonized for less (though prestige in Africa was huge and it's much closer, so perhaps not the best comparison).
As for the latter, Korea being a 'Japan/China' story has only been true for less than 3/4th of a century in the past 1500+ years. Japan, after the fall of Baekje in the 5th century, maintained only trade relations with Korea and played no role politically or socially until the latter half of the 19th century, the Imjin War of the 17th century being the sole exception in 1500 years. Culture flowed from China to Korea to Japan, Japanese culture did not flow back to the mainland to a notable extent until the modern era.
It's like saying Vietnam is a China/Thai/Cambodian story.
You can't 'colonize' an East Asian Nation. Japan pretty much fulfills your criteria, as that was the only way it was gonna happen. East Asia, unlike most of Africa and the Americas, has always been pretty densely populated. There's no place to settle people to run the colony and to provide the basis for an upper class without some level of ethnic cleansing, which would immediately get red exclamation marks from the rest of East Asia.
That said about Vietnam, Vietnam's population was at least comparable to Korea's by the 1860s and it was still taken piecemeal by the French. Also a nation in the Sinosphere and a Qing tributary, also far from Europe, also not ethnically cleansed to any large extent.
Though Vietnam does have a great deal more regionalism due to geography, its main population centers were separated by a great deal of distance and so could be detached without too much trouble due to existing bureaucratic systems, and it was more tolerant of Christian missionaries than Joseon Korea (which played its role from what I've heard).