We've got blown-in insulation in our attic, and I'm tired of everytime
someone goes up into the attic having little bits of insulation
falling out and being tracked all over the house.

If you're using the attic for storage then put down a wood floor. Cut
sheets of plywood into pieces that fit up through the access door and nail
'em down to the joists. This will give you more storage, keep the
insulation in place and avoid the risk of someone accidentally stepping on
the ceiling and falling through it. It doesn't have to be anything special,
any old grade of plywood would work, even recycled from other purposes. But
thicker sheets will take weight better. Just be sure to anchor it reliably,
you don't want it moving around or bowing when someone steps on it.

If you've got a lot of insulation up there and it's deeper than the current
top of the joists then you'll have to add to them to provide a secure
mounting for the flooring. This is messy work, you might want to find a
local handyman to do it.

Is it safe to loosely cover parts of the insulation with plastic
sheeting to keep the insulation more in place?

Plastic generally traps moisture. It would seem like a bad idea to put that
up in an attic. If you can't put down wood then something more breathable
would be better than plastic. Not sure what the building code allows
though. You could, I suppose, just staple down some cardboard or heavy
craft paper. That's sort of half-assed, but if all you're doing is storing
the christmas ornaments it might be enough.

Bear in mind that whatever you put down might at some point need to come
back up. To add insulation, fix a leak or wiring, etc. So don't go putting
down something that'll be a real pain in the ass to remove later on.

-Bill Kearney