It's actually pretty easy to get a direct download link for your Dropbox files. All you have to do is copy the normal share link and tweak it a bit.
How to share a direct download link from Dropbox

First, you need the normal sharing link from Dropbox. Go to Dropbox.com, find your file, and click the Copy link button that appears when you hover over it. Or, on your desktop, right-click on the file, and select Copy Dropbox Link.
Either way, you'll have a link like:
https://www.dropbox.com/s/hriinb9w3a2107m/iPad%20intro.pdf
To get a direct download link, replace the
www.dropbox.com
withdl.dropboxusercontent.com
, which will give you a link like:https://dl.dropboxusercontent.com/s/hriinb9w3a2107m/iPad%20intro.pdf
Copy that link and paste it in your browser, and it should download the file directly. Share that link with your friends and colleagues and they can do the same.
Automatically generate a Dropbox direct download link with Zapier
Another option is to use Zapier's Dropbox integrations to automatically copy and share a direct download link to your new Dropbox files.
Say you want to email a direct link to your colleagues whenever you add a file to your project folder in Dropbox. Just make a Zap that watches Dropbox for new files, then add a Gmail action to send the message. In your email Body box, select Direct Media Link to include a direct download link to your Dropbox file.

You could use that to log your share links in a spreadsheet, post them to your team's Slack project, use them on social networks, and much more. Here are some automations to help you get started:
Send emails via Gmail with new files added to Dropbox
Get notified in Slack when new files are added to Dropbox
And that's how to get a direct download link from Dropbox to share with your colleagues and friends, so they can download your file without having to open the Dropbox website.