I don't know how Dropbox does it, but usually the browser sends a User-Agent string which tells the web server which brower you use, its version and the operating system. Here's an example:
User-Agent: Mozilla/5.0 (X11; Linux x86_64) AppleWebKit/537.22 (KHTML, like Gecko) Chrome/25.0.1364.160 Safari/537.22
You can check what your browser sends e.g. with
netcat. Open up a terminal and open a TCP port:
nc -l -p 3456
Then point your browser to
localhost:3456 and check the result in the terminal. You could edit the User-Agent string to a typical one from another operating system and visit the Dropbox page, if the download changes then you have confirmed that Dropbox uses the User-Agent string to determine the OS.