I recently discovered the Custom Basemap Style Editor and enjoy it’s functionality. If I make maps in the editor do I own the content produced? If so, can I use it for commercial purposes?
One suggestion is to review the Esri / Online terms of use: Online terms of use—ArcGIS Online Help | Documentation
If you are redesigning maps via the Editor, you will not own the content since that's the data Esri licenses and makes available for use (see TOU). It will be your cartographic design.
There are other subtopics about using the maps for commercial purposes on the page linked above, with information on static map use as well as transaction limits.