You can save files with the visible connections list by using the File/Save menu, File/Save As menu, Ctrl+S shortcut or the Save toolbar button. Save and Save As are identical on the first save. Afterwards Save just appends the current list of connections to the previous file, while Save As allows you to either replace the file or save using a new file name. All the column information netactview currently has is saved regardless of the columns you set to see in the View menu.
By default netactview saves a formatted text file. You can change that with the save dialog file filter selector. There are 2 file formats that can be saved: