Which hotel is good for stay in korea with a baby? Any recommendation?

Not sure which area are you asking for, you can consider Metro Hotel in Myeongdong. They offer family friendly facilities such as child equipment rental (Strollers, Bouncer, Baby crib, Waterproof mattress cover for bed wetting), Child safety kits (outlet covers, toilet trainer seat, night light, baby chair, booster) and some other stuff. You can refer to this link for more information: https://www.metrohotel.co.kr:11031/specialoffer/family.asp