Every practical way to make this trip, cheapest and simplest first.
Follow the signs down to the Airport Railroad (AREX) station in the basement of your terminal and take the All-Stop train to Hongik University Station. Change there to Seoul Subway Line 2 and ride to Samseong Station. Leave by Exit 5 or 6 and walk through the underground passage under Millennium Plaza, which brings you straight into COEX Mall without going outside.
about 100 min · about KRW 4,650 from Terminal 1 or KRW 5,250 from Terminal 2, on a transport card · The cheapest way in, and fine if you are travelling light.
Take bus 6103, the non-stop airport limousine to Samseong. At Terminal 1 the stop is 7A-3 on the 1st floor arrivals level; at Terminal 2 it is stop 21 on basement level 1. Stay on to the last stop, the City Airport terminal at COEX, which puts you on the 2nd floor right beside Samseong Station Exit 5.
about 65 min from Terminal 1, about 85 min from Terminal 2 · about KRW 18,000 · Arriving with luggage or exhibition material, with no stairs and no changes.
Take the AREX All-Stop train as far as Gimpo Airport Station, then change to Seoul Subway Line 9 towards Bongeunsa Station. Leave by Exit 7 and use the covered passage through ASEM Plaza into COEX Mall. Make sure you board a regular Line 9 train, because the express trains do not stop at Bongeunsa.
about 110 min · about KRW 4,350 from Terminal 1 on a transport card, plus the Line 9 leg on the same card · Handy if your hotel is on the Line 9 side of COEX.
Taxi ranks are directly outside arrivals at both terminals; you do not need to book. Show the driver the venue name in Korean, 코엑스, or the address 서울 강남구 영동대로 513. Expressway tolls, the out-of-city surcharge and any late-night surcharge are added on top of the meter.
about 60 min · about KRW 60,000 in a standard taxi, about KRW 90,000 in a black deluxe taxi, plus tolls · Late arrivals, or two or three people splitting the fare.
The last 6103 bus of the day leaves Terminal 1 at about 23:30 and Terminal 2 at about 23:10 — land later than that and a taxi is your only option.
Incheon Airport (ICN) → COEX, Samseong, Seoul →
Gimpo Airport (GMP) → COEX, Seoul →