Add a custom body class to WordPress for the current browser being used. No external frameworks or plugins.

browser-body-class.php

<?php

function custom_body_classes($classes)
{
    // the list of WordPress global browser checks
    // https://codex.wordpress.org/Global_Variables#Browser_Detection_Booleans
    $browsers = ['is_iphone', 'is_chrome', 'is_safari', 'is_NS4', 'is_opera', 'is_macIE', 'is_winIE', 'is_gecko', 'is_lynx', 'is_IE', 'is_edge'];

    // check the globals to see if the browser is in there and return a string with the match
    $classes[] = join(' ', array_filter($browsers, function ($browser) {
        return $GLOBALS[$browser];
    }));

    return $classes;
}

// call the filter for the body class
add_filter('body_class', 'custom_body_classes');