Yes and no. As long as you have a USB cord that works with your phone, then you can hook it up to your computer and transfer the files. You can transfer them, but you are definitely going to need the phone charged to do so.
If it only holds a little charge at a time, and then dies, you will be doing the transferring for a while. I say that because it will have to charge often, so it will take longer if you have a lot of things to transfer. Unless, of course, it can hold a charge long enough to move the files.
You also have the option of purchasing a new battery (they are usually pretty cheap) and that would give you more charge time.