Korea Expose published an interesting article about the forgotten history of the above set of steps in Haebangchon. Especially interesting was the interview with a local woman who remembers going to the Shinto shrine which used to be at the top of these stairs. The article risks confusing this shrine with another, however.
The actual Gyeongseong [Keijo, or Seoul] Shrine was built on the slopes of Namsan south of what is now Myeongdong Station in 1898; a few stone remains can be seen behind Sungui Women's University. It stood not so far from the original Government General Building (built in 1907 before moving, famously, to the large building that stood behind Gwanghwamun until 1996). Also nearby was the Japanese ambassador's residence (built in 1893, before another was built on what is now Yongsan Garrison in 1909, before the final one was built in 1937 on the location of today's Blue House). Photos of all of these can be seen here.
The more famous Chosen Shrine was built in 1925 and almost became the location of a new national assembly in the early 1960s; it now has an Ahn Jung-geun museum and other monuments to Korean independence fighters. There was also a military-related shrine on what is now Yongsan Garrison (I've never seen any photos of it) as well as other smaller ones throughout the Japanese parts of the city and throughout the country. These did not survive past 1945.
The shrine in the Korea Expose article was the 경성호국신사 (more photos can be seen here). If we follow Norma Field's translation of 호국신사 (in In the Realm of a Dying Emperor: A Portrait of Japan at Century's End), this would be the Gyeongseong Defense-of-the-Nation Shrine. She writes that in 1939 a directive stated that each prefecture in Japan was to have one official such shrine. The souls of dead soldiers were to be enshrined there, and if this sounds familiar, it might be because these were essentially local branches of Yasukuni Shrine. Seoul's was built in 1943, and I'm not sure if Korea had only one such shrine in Seoul, or more than one (though I'd lean towards just one). They would have been used not just for enshrinement of Koreans (who were only being used by the Japanese military in small numbers as volunteers or POW guards up until 1944) but for Japanese who were living in Korea.
At any rate, it would be a shame to see those stairs disappear, which the article states is a possibility. Surely if some of the secondary stairways related to the main shrine on Namsan (now standing near memorials to independence fighters) can be allowed to remain, these can as well.