Blogger does not provide you an easy way to resize your header image. Imagine how troublesome it is to get your image's dimensions to be perfect before you can actually upload it to Blogger. But usually we face a less-than-ideal case. So we'll be left with only one option, to resize our original image and then re-upload them.
In this tutorial, I will share with you 2 methods on how you can resize your image directly by tweaking some code without having to resize your original image and reupload this image to Blogger. Of course, you can always opt to resize your image (using MS Paint, Photoshop or online image resizing websites like http://www.resizeyourimage.com/) and then reupload your image to Blogger. But it's always more convenient to just change a number or two and see an instant change in your header's image.
UPDATE: If you're a beginner, check this link out instead.
I will be using Simple Template by Josh Peterson in this tutorial. It should work with other built-in and third party templates as well. If it doesn't, the idea still remains the same. It's hard to provide one generalised solution to all the templates out there.
This tweak will not work if you choose to display your title and description over your header image. This is because when your title is displayed, technically your header image acts as a background (instead of an image object). It is not possible to conveniently style the size of a background image, although there is a CSS3 property that can be used (background-size). Unfortunately, at the time this tutorial was written, most browsers do not support CSS3 yet. So you'd have to hide your title and description to allow this tweak to work for now. To hide your title and description, go to Page Elements/Layout - Edit on Header - under Placement, choose 'Instead of Title and Description'.