Como fazer funcionar o atributo required no IE8 e Safari?

Não dá... de forma nativa, mas podemos utilizar o Webshims

Exemplo:

<script src="//cdnjs.cloudflare.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<script src="//cdnjs.cloudflare.com/ajax/libs/modernizr/2.8.3/modernizr.min.js"></script>
<script src="//cdnjs.cloudflare.com/ajax/libs/webshim/1.14.2/minified/polyfiller.js"></script>

<script>jQuery.webshims.polyfill('forms');</script>

<form>
    <input type="text" required value="" />
    <input type="submit" value="Submit" />
</form>

Fontes:
https://developer.mozilla.org/pt-BR/docs/Web/HTML/Element/Input
http://www.html5rocks.com/en/tutorials/forms/constraintvalidation/