The XNAT Desktop Client has a data transfer management panel for both uploads and downloads, allowing you to see the progress of the actions that you initiated. You can also pause, resume, or cancel those actions.
Each transfer log is specific to the user account and XNAT instance that you are logged in with.
The Desktop Client allows you to pause and resume transfers individually, if you would like to prioritize one transfer over another over limited network bandwidth, or pause all transfers. Controls for affecting all transfers are located at the top of the Transfer Monitor page. Controls for affecting an individual transfer are located in the details modal for that transfer. (Click "Details" to view the modal.)
You can cancel a transfer if you like, and remove all canceled transfers from your queue. However, you can also restart a canceled download if you like.
If you want to keep a log of all your data transfers, feel free. However, if you find the long list to be unwieldy, you can clear completed transfers from this window by using the "Clear Completed Transfers" button at the top of the page.