How to use WordPress 3.8 back-end CSS in front-end?

I believe you’re asking how to include the admin styles in your front end theme? If so, you simply need to either enqueue ‘wp-admin’ on it’s own or list it as a dependency for your theme/plugin stylesheet. Here are examples of both methods:

By itself:

wp_enqueue_style( 'wp-admin' );

As a dependency:

wp_enqueue_style( 'my-theme-styles', 'path-to-my-css.css', array( 'wp-admin' ), '1.0.0' );

There are also several individual parts of the admin css that can be enqueued separately as well:

'dashicons'
'admin-bar'
'thickbox'
'wp-admin'
'buttons'
'colors'

Although enqueueing any of these individually may have less than ideal results as I don’t believe they are listed as dependent on eachother.