Can you try connecting your GPS, wait for it to load, go to it's folder and drag an item from that folder, like a Waypoint and move it to My Collection at the top of BaseCamp? Does this work for you?
First, ensure you have Custom POIs selected in Maps, Manage Custom POIs. If they don't show there you can't do anything.
If they do show then you can view/search for them. You can then right click them and save them as waypoints. Note you'll probably need to have detail set to Highest
What I have done in the past is to copy the POI file from my device to my computer's harddrive and then convert the file from gpi format to gpx format using one of many gpi to gpx converters available on the internet. Then import into basecamp.
I'm now confused. Are you talking about POIs that POI loader has placed on your nuvi or .gpi files that you've put there yourself? If neither and you're talking about the POIs that are part of the Garmin map then you can't transfer them anywhere as they're not separate files.