Things don't "pick up" radiation just because they are exposed to it. During the aftermath of a nuclear blast or accident like Chernobyl, there is a lot of dust and other particulate containing radioactive material that can easily spread contamination to food, water, through skin etc, and that has led to the misperception that if something is exposed to radiation it can then "pass it on" to other things.
It's like shining a really bright flashlight on your balloon, and then later wondering if the light from the flashlight will affect you if you keep the balloon at home - it just doesn't work that way.
Unless you balloon encounters a cloud of radioactive dust in the atmosphere (won't happen) there's nothing to worry about no matter how much it gets exposed to.
If it makes you feel any better, I scanned my module and parachute from my last flight with a geiger counter and nothing showed up except the normal background radiation we get every day which is about 1/50th of what airline crews get exposed to every day they work with (apparently) no measurable health effects. It's also about half of what comes out of your average granite countertop.
If you're worried about radiation sources, go buy yourself a handheld geiger counter from ebay. You'll be amazed at how un-radioactive things we worry about like water, ocean fish, and the air actually are. IT's also fun to take it to antique stores and look for uranium glass and especially orange fiestaware plates which are actually quite hot.