Most people do that as they create the image by using the add text feature of their image software.

I wouldn’t recommend doing it in code automatically because you might have occasion to share a post with images, have a guest post with images or borrow an image from creative commons… you sure don’t want to put a copyright notice on someone else’s property.

Here’s a plugin if you really want to do this through Image Watermark WP

I’ve never used the above plugin.

Also, while we are talking about plugins and images you might find ImageInject real handy. This will search for images released under the creative commons and place a little Copyright ‘blurb’ accrediting the CC with the image if you use it.

It’s pretty handy. Images are a good thing.