Is it possible to upload two different versions of a site (desktop version and mobile version) and have netlify point to either one depending on the device they someone is using?
The top banner of my website looks abit weird cropped when viewing on a mobile. I have used mobirise to build the website.
Personally, I don’t think that’s the way you’d best tackle this issue!
There’s a great discussion over at Smashing which explores media queries and responsive design. In short, there’s no reason why the banner can’t resize. 
@anthdidj Short answer: Yes.
Longer answer, @Scott is correct.
To direct visitors to the correct version for their device you need to do some kind of browser sniffing, which is problematic at best, and at worst creates a situation where you need to keep on top of changes to browsers and devices to make certain each visitor gets the content you think he should get.
Just create your website using responsive CSS and you should be fine.