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".
Windows Explorer view settings should be set to "List" or "Details" for best results in WebDrive.
If the issue still remains, Windows Explorer or an Explorer extension is requesting these files to be downloaded.