I can't speak to something like Android, but nothing will get downloaded to your desktop or laptop as long as you don't click on anything in the new tabs that the site tries to open. But if you're going to visit a site like Mixdrop where you know tabs / new windows open, you can always set your browser to block those. Pretty much all browsers provide this option, so there should be no reason for them to show up in the first place.
It's usually under something like a Privacy section, Site Settings, or similar within the browser settings. You'll want to look for something like "Tracker and Ad Blocking", "Content", "Third-Party", or things along those lines. It varies from browser to browser, so you can either search within the browser settings screen or do a web search to see instructions.
And even if you forget or can't figure out how to change those settings, you still aren't forced to click on anything. Just have
Ctrl +
W (or
⌘ Command +
W) ready and when a new tab opens close it immediately. Click
Download again and close the new tab again. Just repeat until when you click
Download the file actually begins downloading. (And if any final new tab opened, obviously
Ctrl +
W it again so it goes away.) I find I need to close probably between 5 and 7 tabs that open before the file starts downloading.
I do think the staff here should revisit whether Mixdrop should be an acceptable mirror site. Still, there are other mirrors to use as well, so just steer clear of Mixdrop if you're worried.