If you need to take your gear on a flight, read this USPA and TSA travel advisory. It is written for skydiving rigs, but applies to BASE rigs as well.
The TSA advices you to take your rig as a carry-on instead of checking it in. The main reasoning is that the TSA requires the owner of the rig to be there if they decide to open the container. If your rig is checked in, it’s going to be a lot more hassle to find you.
Skydiving rigs can be a bigger problem than BASE rigs because of the AAD which can show up as a blackbox on their X-ray machines.
Another good reason to have your rig as carry-on is that checked-in luggage can end up missing more easily. Usually it arrives later that day or the next day, but sometimes it just disappears completely. The most important thing to have at your destination is your rig. You can always buy a pair of clean undies and borrow a helmet from somebody. Finding gear to jump in on such short notice is a lot harder.
It is highly recommended to have your rig in the stashbag with nothing else in it. Be sure to take the hookknife off and put that with your checked luggage. While there are reports of people having taken hookknives as carry-on, there is no need to raise suspicion.
I have checked in rigs with my other luggage and I’ve taken them with me as carry-ons; neither has ever posed a problem. BASE-rigs are really no big deal, they’re just a backpack with some fabric in them. I think that taking a backpack with some rope and some bedsheets would raise the same suspicion: none.
I’ve passed airport security with carry-on skydiving and base rigs at least eighteen times and never even had anybody ask me about it.