WebDrive isn't automatically downloading the files - it's actually Windows Explorer or another process downloading the files anytime something accesses them. Most likely, there are programs running in the background, such as anti virus real-time scanning, that may be accessing these files. WebDrive will NEVER download a file on it's own. Explorer column handlers or thumbnail viewers do this.
You can try the following to prevent downloading by Explorer:
In WebDrive, under App Settings, make sure you have enabled "Disable shell column handlers". Also, click on the site profile of interest and go to Site Properties --> Basic File Settings, and make sure you have enabled "Disable explorer icon handler to prevent file download".