The card is the "Air 1.5" from ShinhanCard (Link (in Korean)). The card does not really have any other perks like priority pass, but it costs just 45,000 KRW a year and in return, you get 1.5 miles per 1,000 spent domestically in Korea and 3.0 miles for any foreign spending (the extra foreign spending bonus is limited up to 2,000 miles per month).

A coworker recently told me that he got a new Citi Card which actually offers 1.6 miles per 1,000 KRW and has priority pass. It also seems to let you transfer miles to some other airline programs like SQ, albeit at less favorable rates. The downside is it costs 150,000 KRW a year and does not have any additional bonus for foreign spending.

If I were shopping for a new card I might actually consider the Citi Bank one, but for now I'm pretty happy with the Shinhan Card one.

Edit: It seems the Citi Bank link doesn't work. The card is called "Citi New PremierMiles Asiana" (씨티 NEW 프리미어마일 아시아나)