How to strip non-alphanumeric characters, convert spaces to dashes, uppercase to lowercase in this context

You could use sanitize_html_class() or sanitize_title_with_dashes(). Both functions do almost the same. But note that much more characters a valid in CSS class names.

Examples: ✂ ✉ ✔✽ ✾ ✿ ❀ ❉ ❤ ❦ ☂ ☔ ☎ ☘ ♫ ♻ ⚥

Update

I recommend to encapsulate the code in a prefixed function to avoid collisions with poorly written plugin using the same names as you do.

Example (not tested):

function wpse_48986_custom_fields_to_classes( $field_name )
{
    $values = get_field( $field_name );

    if ( is_string( $values ) )
    {
        return trim( sanitize_html_class( $values ) );
    }

    if ( is_array( $values ) )
    {
        // Prepare the array values.
        $values = array_map( 'sanitize_html_class', $values );
        $values = array_map( 'trim', $values );
        return implode( ' ', $values );
    }

    // We didn't get a string or an array.
    return '';
}

And when you need the CSS classes somewhere just write:

print wpse_48986_custom_fields_to_classes( 'field_name' );

I couldn’t test the code code: I don’t have the plugin installed, and my main computer is broken, so there may be errors. Feel free to edit this post and fix the broken code. 🙂