Insulation picture https://imgur.com/gallery/gpP6RE7

Are you sure that is safe to be around? Some types of insulation are quite dangerous for the lungs.

Never thought playing ball in basement is good idea. When a ball busts a pipe you might find yourself scrambling to shut off the water to prevent a full on flood. Put up some temporary plywood sheets in places of concern. You can suspend sheets lower if need be for clearance of ducts etc. You can extend 2x4s down to hang sheets.

Cheapest would be to tack up sheets or plastic. That would hold all the insulation in and prevent anything from getting knocked around. It's possible that this could create a rodent hideaway but minimal risk. I wouldn't invest in sheet goods since you aren't planning on living here long term.

That stuff is very unhealthy for lungs, hands eyes. Def tyvek with staples.then add scant strapping 9’ roll of tyvek on down.

Staple something to the bottom of the joists to close it all in. House wrap, paper, tarps. Could also do thin plywood or masonite panels.

From what I understood, this is a soundproofing insulation made from cotton fibres and isn't harmful to be around.

Dodgeballs and nerf darts are made of foam. Even if Chuck Norris was playing, I don't think our pipes would be in danger.

That’s Roxall insulation. Cool Canadian product. My wife used to serve the founder. As someone else suggested, tyvec or vapour barrier. 6ml vapour barrier will likely be the cheapest solution.

Are you serious?

Check your fire code. This sounds like a violation.

I’d do a drop ceiling. Pretty easy to do and not very expensive. It’ll also add a little more sound dampening.

If you really want to go cheap and easy, you can staple some plastic sheeting to the ceiling instead. It’ll serve it’s purpose but be pretty ugly. Hang it on the walls too, if you really want to get a murder room look.

I put a plastic 3mil covering over mine to keep the partials from falling on everything. Not saying this is the right answer but it works and its inexpensive. I do plan to put lighting in at some point and I will probably drywall over it due to the height of the ceiling.

Even cellulose (basically paper fibers) is bad to breathe in. Mineral wool (which is what it looks like) isn’t good for the lungs. A little here and there isn’t that big of a deal, but I wouldn’t make a habit of breathing it in. Especially for kids with developing lungs.

How much do you feel like spending?

Where i live i can get 4x8 sheets of 5mm plywood "underlayment" for $13 per and they'd be super easy to put up with a brad nailer or castle stapler and from there you could paint 'em. Or get fancy and add some battens to cover the seams then stain it all.

Pretty much all insulation products are harmful if inhaled, and every time a ball hits that stuff, what you are not seeing is small particulate being broken loose and sent airborne… to drift down over your kiddies heads. Probably not life threatening in limited exposure, but definitely not beneficial!

If it were me, I’d screw up the cheapest sheet good I could lay my hands on, or as others have suggested, staple up some 6mm poly and tape the seams with tyvek tape (the red plastic-y stuff). 6mm is pretty robust, so it should stand up to some abuse, but if you go too thin somebody is bound to poke holes in it eventually.