Currently the maximum width of images is 630px (all.css →
.post-text img rule). With a 15:9 aspect ratio this gives a height of 1050px; at 16:9 it's 1120px.
Assuming SE allows this level of customisation for the individual sites, the simplest solution would be to simply decrease the default max-width of images. 400px seems to be a reasonable compromose - not too big, but text is still legible. If you want to try it out for yourself, most modern browsers should have integrated developer tools that can modify stylesheets on the fly, launched by pressing F12 (in some browsers you can also right-click on a screenshot and select "Inspect").
Resize options would be nice to have as well, but if the majority of screenshots are going to be in portrait rather than landscape (which is the case with smartphones), it seems to me that an automatic solution that doesn't require every image to be resized individually would be a better option.
Just my two cents.
Alternatively, images can already be resized, you'll simply have to use HTML instead of Markdown. So instead of this:
![enter image description here]
Use something like this:
<img src="http://i.stack.imgur.com/foobar" width="400">
Example: What exactly is needed to activate Cortana? (just edited the question to make the screenshot smaller)
Further, if the screenshot contains a lot of detail, you can also make it link to the full-size version (with an optional footnote at the end):
<a href="http://i.stack.imgur.com/foobar"><img src="http://i.stack.imgur.com/foobar" width="400"></a>
<sup>(Click for full-size version)</sup>
This is of course probably a bit much to expect from new users, but at least it's something that can be done right now about posts where the size of the screenshots is especially problematic.