there's a number of steps needed to achieve this:
1 - Make a copy of the line dataset
2 - Create 2 fields in your line dataset and do a "calculate geometry" to calculate the length of each line in one of these fields, and add a unique name or number to each line in the other field.
3 - cut each line where it intersects a polygon.
4 - Add another field to the line dataset and calculate the new length.
5 - Now you have the old length and the new length, so you will be able to calculate which lines have 60% inside the polygons. Select them.
6 - Export these selected lines (which will be segments of the original lines). You can now do a spatial selection on your copy of the line dataset (the copy you made in step 1) to select the full lines.