The only way currently to know if a Box Note has an Active Shared Link is to go to the folder which contains it (in the regular Box.com UI, not the Box Notes UI), and look for the "Active Shared Link" icon on the row for the .boxnote file.

I suggest adding this same icon to the sidebar box for the note in the Box Notes UI (i.e. the box you click to load the note in the main right-hand pane).

See: https://community.box.com/t5/custom/page/page-id/BoxViewTicketDetail?ticket_id=1853249