How to download documents using content server api's

My guess is you will have to write a program in some language that understands how to call the REST API

If you are new to REST API or CS observe CS in the SmartUI that is the REST API in cation behind the scenes, Then download Postman and create simple examples and design a program.

Alternatively you can buy tools that other vendors have already coded

developer.opentext.com and CS API's are the first LOB.