I have not found a way to export/download just the supplier comments - but I do use the following to export all responses, including the comments.
In OD Sourcing Pro I use the table options menu - Export All Rows option for the Monitor-Content tab of an event. You will want to make sure all suppliers are checked in table options - participants filter so that you export all participants. This exports all responses including the comments to MS Excel, the only thing it doesn't export is the attachment file name if a supplier added an attachment to the comment. I format in excel and then copy and do a little formatting in Word to make review by stakeholders cleaner. Most of the time I have suppliers that provide extensive responses that do not view or print well in excel - so I've found using MS Word useful.
This is what I have found to be useful - but am interested in learning how other companies are doing it too.
Hope this helps!