Is the linked image in your CSS file? In which case, you can't add alternative text I don't think.

If it's coded somewhere else, edit the file/widget/whatever, find the image code you want, and add

alt="your description here"

to the image code. Ie.
<img src="whatever URL you're using for the image" alt="your description here" />

I hope that's what you meant, I'm not entirely sure if that's what you were asking.

Try this

$File->desc
$File->alt
$File->title

Thanks guys, thought I deleted this post since I got it. Your reply is correct Sam.