If you have family coming with you to Korea who will not be working, you need to obtain an F3 visa (dependent family visa) for that family member. These visas are for spouses or children of an E2 visa holder.
An F3 holder is allowed to remain in Korea the duration of the contracted visa but is not allowed to work.
You can arrange for this visa at the same time as you apply for your E2 visa. Submit all required documents together. Or you can wait and have your family enter on a tourist visa and then change it to an F3 visa at your local immigration office in Korea after you arrive.
Here is what you need:
A passport valid for the duration of the visa
1 passport-sized photo
The original marriage certificate/ or birth certificate for children
A copy of the front and back of the E2 visa holders Alien Card (*when applying at Korean Immigration)
Proof of the E2 visa (i.e. Visa Issuance Number)
Visa fee (please contact your consulate for the exact rate)
Visa Application Form