How to save custom theme settings in a subtheme?
I ran into the same issue. After doing an analysis I found the following:


THEME_form_system_theme_settings_alter is called for the base theme and then for the sub theme. It's not really a hook as they say - "Implements hook_form_FORM_ID_alter().".
The base theme uses things like theme_get_setting('asdf_asdf', 'responsive') to get the default values for the settings form.
The settings you're saving are saved for the sub theme.
In templates settings are returned also using theme_get_setting('asdf_asdf', 'responsive').


Because of #2 and #3 the settings appear to not be saved.

Maybe they should have used theme_get_setting('asdf_asdf'). So it will default to the current theme. It can be used in both cases: #2 and #4. It would have worked fine.

For the sake of not altering the theme and because of #4, I would remove the extra settings in SUB_THEME_form_system_theme_settings_alter so you can modify the settings only in the base theme configuration page.