Although many correct answers have already been given, the classical (read non-DOM) approach would be like this:
document.forms['yourform']['yourtextarea'].value = 'yourvalue';
where in the HTML your textarea is nested somewhere in a form like this:
<form name="yourform">
<textarea name="yourtextarea" rows="10" cols="60"></textarea>
</form>
And as it happens, that would work with Netscape Navigator 4 and Internet Explorer 3 too. And, not unimportant, Internet Explorer on mobile devices.