You can create an RSS feed of any of your put.io folders (even your root Your Files folder) to automate downloads from put.io on various platforms. Here are the steps:
First, select the folder from which you want to create a feed. Then, select RSS Feed from the Actions dropdown menu. A popup window will open, allowing you to filter the feed contents by type. Click the button that matches your preference (if unsure, select All). The URL will then appear at the bottom of the popup window. You can now use this URL in your RSS client or Synology Download Station.
One problem that you might run into is that this URL does not contain any authentication information, so will prompt for a username/password when used on a different machine or browser. One simple way to add your account information to the URL is to embed it like this https://username:password@api.put.io/rss/all/12345678. So if your username is "awesome" and your password is "secret", then your URL would be https://awesome:secret@api.put.io/rss/all/12345678.
Another way is to create a new OAuth app from https://app.put.io/settings/account/oauth/apps, get the token from there, and append oauth_token param to RSS URL: https://api.put.io/rss/all/123456?oauth_token=XXXXXXXXX.
Please be careful when using these types of links, though, since they will include your credentials in the open. We advise you to only use these types of links in your own clients.