Yes, for a broader and stronger protection in South Korea, we recommend registering your trademark in local characters.
If the trademark is registered only in its original version (Latin characters), the protection does not always properly protect its equivalent in the local language. This means that a third party could use or register the same trademark (or a similar one) in local characters.
In additional to the legal benefits, the registration and use of the trademark in local characters can also have commercial benefits. The public in South Korea will recognize your brand more easily if they are able to read and correctly pronounce the mark.
Registering your trademark in its original version, as well as local transliteration/translation, will provide a greater protection from any possible infringements.