Thanks for your input.
I am aware of how Firefox manages downloads. However, as I wrote in my original post:
"When I try to download in my browser, it always stops downloading after about 38MB (the file is 800MB)."
That is, the download just stops downloading. I've repeated this a dozen times. I am not stopping it. It simply stops downloading.
The problem with using Filezilla (which I do use and like very much) is finding the backup file. Ken's suggestion will likely work but now I need to install the ad-hoc query plugin and do some sql queries. All of this is possible, it's just not nearly as easy as I would prefer it to be.
Once I get out from under my current load, I will take care of this. Until then, I will hope that the hosting server does not crash.