What else can I say but: How can I wash chopsticks in the dishwasher without them falling?
Line the bottom of your utensils basket with a wire mesh. You can buy one for drains for about $1 and flatten it out until it fits.
Lie them down in the top rack next to or under the glasses and mugs you have up there. Mine get clean there no problem.
Hold them to the rack with a wooden clothes pin. The spring loaded ones work more easily, but the old clamp on kind will also do the job.
How about using a rubber band to join them thick end to thick end and lay them flat where they cannot fall through the grid and are too long to move around much.
Or place the joined pair into the cutlery box, they cannot slip through the holes.
Not having had this problem as chopsticks are quick to hand wash and the disposables go in the bin. I have not tried hard to find solution. However the simplest way would likely be to use a clothespeg to clip them onto the structure or to each other. This should make they stay in place and is simple, cheap and repeatable with no fuss or bother. A stainless steel string may be needed for regular use to prevent rust stains.
In my dishwasher, there is a basket for the utensils. If the holes of the basket are large enough for the chopsticks to fall through, then slide them into the basket horizontally. If the holes are too small, then you can just stick them in the basket like any other utensil.
If that doesn't work, I'd probably just get a brand new sponge dedicated for the purpose and push a dozen chopsticks into it, half a chopstick sticking out on either side, and put the whole thing on the top rack. Alternatively, use something like a net and make a little pouch with a drawstring as a chopstick bag, like some people do for socks in laundry machines.
Do so at your own caution: I bear no responsibility if your sponge catches on fire. =P