The behaviour depends on the browser. In some browsers the dialogue box isn't dismissed until the copying of the streamer down file is complete, in others, the box disappears quickly but the file copying continues in the background: if you refresh the Web-If media browser page in the latter case you will see the file growing. If you cancel before pressing save, the process is cancelled. If you cancel after saving, the save operation will complete in the background. When you select save you should get a pop-up with lots of info about the download: if you just get a dialogue box to input the file name without this info, something has gone wrong: cancel and try again.
It was easier with the older version of iPlayer (V3) as you could see the download progress on screen. Now you only have Web-If to monitor the download. It is fine to save the download with iPlayer still running but I recommend waiting until you are sure all the file has downloaded. If you do stop the download early, when you play the resulting mp4 file, it will seem to be the right length but you won't have all of it: at some point playback will stop. Make sure the save is complete before watching a different iPlayer programme or YouTube clip as this will wipe the streamer down file and mess up your download.