Some handy websites on how to pack a banjo for shipping:
There have been sad stories on the Hangout, about carriers denying insurance claims where a used box is involved, saying this is improper packing. Two luthiers that I have dealt with, use double-corrugated boxes. No, these are not inexpensive, and yes, the additional weight adds to the shipping price. Gruhn Guitars recently was paranoid to the point that if you buy a banjo from them, they ship with the neck detached (this was a few years ago, not sure if this still is their practice).
A problem with packing peanuts, is that they do settle, which might allow the banjo to slide around. If you use them, take the time to ramrod them down. I've made balls of bubble wrap, to secure the instrument case from knocking around inside the box. Use quality packing tape. I once had a banjo shipped to me, arrived with a hole punched in it and a lot of the packing peanuts had dribbled out, but the banjo was OK because the case was braced with wadded butcher paper and other materials that still cushioned and kept the banjo immobilized inside the box.
When shipping, I've taken date-stamped digital photos of the packing process, in case of an insurance claim. I also take pictures of the unpacking process when I'm receiving. Whenever possible, I divert the shipment to a nearby FedEx or UPS site, wait at least 12 hours for the temperature to stabilize, and open the box there so that there's no argument on the condition. So far, so good, but stuff happens, and I figure the prepared are better positioned to fight in an insurance claim if something happens.
Standard disclaimer: YMMV. No guarantees or warranties. Always some risk when shipping a banjo, stuff happens. Good luck!