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Set Sublime Text as the default handler for public.plain-text

defaults write com.apple.LaunchServices LSHandlers -array-add \
'{LSHandlerContentType=public.plain-text;LSHandlerRoleAll=com.sublimetext.3;}'

This answer is for Mavericks and earlier.

For Yosemite and later, use com.apple.LaunchServices/com.apple.launchservices.secure.

defaults write com.apple.LaunchServices/com.apple.launchservices.secure LSHandlers -array-add \
'{LSHandlerContentType=public.plain-text;LSHandlerRoleAll=com.sublimetext.3;}'

Remember to restart to pick up the changes.

To set Sublime Text as the default handler for public.plain-text:

Mavericks (10.9) and earlier

defaults write com.apple.LaunchServices \
    LSHandlers -array-add \
    '{LSHandlerContentType=public.plain-text;LSHandlerRoleAll=com.sublimetext.3;}'

Yosemite (10.10) and later

Use com.apple.LaunchServices/com.apple.launchservices.secure.

defaults write com.apple.LaunchServices/com.apple.launchservices.secure \
    LSHandlers -array-add \
    '{LSHandlerContentType=public.plain-text;LSHandlerRoleAll=com.sublimetext.3;}'

Remember to restart to pick up the changes.

Set Sublime Text as the default handler for public.plain-text

defaults write com.apple.LaunchServices LSHandlers -array-add \
'{LSHandlerContentType=public.plain-text;LSHandlerRoleAll=com.sublimetext.3;}'

This answer is for Mavericks and earlier.
 For Yosemite and later, use com.apple.LaunchServices/com.apple.launchservices.secure.

Set Sublime Text as the default handler for public.plain-text

defaults write com.apple.LaunchServices LSHandlers -array-add \
'{LSHandlerContentType=public.plain-text;LSHandlerRoleAll=com.sublimetext.3;}'

This answer is for Mavericks and earlier.

For Yosemite and later, use com.apple.LaunchServices/com.apple.launchservices.secure.

defaults write com.apple.LaunchServices/com.apple.launchservices.secure LSHandlers -array-add \
'{LSHandlerContentType=public.plain-text;LSHandlerRoleAll=com.sublimetext.3;}'

Remember to restart to pick up the changes.

Set Sublime Text as the default handler for public.plain-text

defaults write com.apple.LaunchServices LSHandlers -array-add \
'{LSHandlerContentType=public.plain-text;LSHandlerRoleAll=com.sublimetext.3;}'

This answer is for Mavericks and earlier.
For Yosemite and later, use com.apple.LaunchServices/com.apple.launchservices.secure.

Set Sublime Text as the default handler for public.plain-text

defaults write com.apple.LaunchServices LSHandlers -array-add \
'{LSHandlerContentType=public.plain-text;LSHandlerRoleAll=com.sublimetext.3;}'

This answer is for Mavericks and earlier.
For Yosemite and later, use com.apple.LaunchServices/com.apple.launchservices.secure.