Save and run gulp.js

It should be run automatically, see https://youtrack.jetbrains.com/issue/WEB-19770

If that's not the case - please describe your project setup

I think I may have things set up incorrectly. I work mainly out of src folder in an AngularJS environment. Currently this is my run configuration. Set to run my gulp default task. 

When you hit Shift+F10 - it starts the Gulp watcher which should be run continuously. Could you please check/observe if it's running that way? Just leave Gulp tool window not minimized, edit some files & try to trigger it: either via Save action, or by focus switching from and into PhpStorm

I hit Shift+F10 and it does run but it gives me this output as well. And then stops running

You can't run Gulp task on Save - on-save actions are not currently supported by the platform, as files are auto-saved by the IDE - see https://youtrack.jetbrains.com/issue/IDEABKL-6722.
If you need to run Gulp on Save All, set it up as file watcher (https://www.jetbrains.com/help/phpstorm/2018.3/using-file-watchers.html), and disable 'Auto-save edited files to trigger the watcher' for this watcher, so that it is triggered on explicit Save/changing focus; or, use Gulp watch tasks to trigger building on Save, like

gulp.task('watch', function () {
    return gulp.watch('src/**/*.js', ['js']);
});