I used carabiners to attach my load tapes to the Kevlar cables, and the cables to the balloon basket. Don't buy carabiners at a local hardware store! These won't cut it! The place I got my stuff from is at: http://www.cmcrescue.com.
I bought 12 Aluminum carabiners rated to 6,744 lbs., and 3 steel ones that are rated to 16,185 lbs each. While these cost a great deal of money (over $200), it is worth it to know that my carabiners will never fail.