Having packed ribbon chutes hundreds of times when racing the jet dragster I can't agree more with the suggestion of having ONE person solely responsible for the chore. So much the better if it is the driver, but absent that, make the driver check the process EVERY time as that is the person that has the most to lose. By the grace of God, my chutes always came out even if they did rip off a couple of times when the car tripped the 1/4 mile lights north of 300MPH.
My job was easier since I had no salt build-up to deal with, only the occasional chaffing when the driver was late on the brakes and dragged the laundry on the pavement. Others have warned about packing the chute when wet and you want to NEVER do that. Also, understand that things wet with a saline solution will take a lot longer to dry. Some racers have a second chute set (mostly cost prohibitive) and others have taken to assigning a crew member to shake out salt then spray the chute/riser/lanyards with clean water at the end of the run/day allowing it to dry fully before packing. Yeah, it is more work, but some get a greater sense of safety in doing so. You will learn your levels of comfort as you go along. While on the salt, if you happen to run into Tom Burkland or Rex Svoboda, ask them to take a look at your set-up - both are experts at this process and always happy to lend a hand.
As starter Bill Taylor would say, "Stay safe, go fast." Always in that order.