I am trying to erase any polygons that do not have a line intersection through them and am unsure of what tool to use using ArcMap 10.1
To remove polygons that don't have an intersection with lines:
Open ArcMap and add the polygons and lines to the table of contents.
Locate the Select by Locations dialog and select the ones that do intersect.
Switch the selection by going to the selection in the table of contents
Right click on the layer name and select 'switch selection'.
Start editing and press delete:
Don't forget to save edits.