I have no idea as to how much the Korean embassy will charge for authentication of a birth certificate.
You need to contact the embassy directly to find out.
Their contact details are as listed below:
Korean Embassy in Makati City, Philippines Embassy of Korea (Republic) in Manila, Philippines 18th Floor, The Pacific Star Building Sen. Gil Puyat corner Makati Avenue Makati City, 1226 Metro Manila, Philippines City: Makati City Phone: (+63) 2 811-6139 to 44 Fax: (+63) 2 811-6148 Web Site: http://embassy_philippines.mofat.go.kr Office Hours: 08.30-12.00 and 13.30-17.30