The traditional solution on bicycles is metal wire basket.
Metal is tougher than plastic and mesh instead of sheet metal makes it lightweight enough.
If you need something waterproof, put your things in waterproof bag or get a more expensive bike-specific rack box.
bicycles.stackexchange.com/users/10595/ojs
Wood is an often-overlooked material.
Its pliable and forgiving of vibration.
Plus its available in heaps of different profiles and weights.
You can construct the size of box you want to the millimetre, not requiring to make use of whatever you find.
For added durability, painting the wood allows you to express yourself, or a good location to attach reflectors and lights.
bicycles.stackexchange.com/users/19705/Criggie
The first container looks to be really thin and cheap, no wonder why it did not last.
The second container looks better, but, IMO, will not last long either.
How is the container attached on your bike ?
You could place a piece of rubber between the container and whatever holds it.
bicycles.stackexchange.com/users/13517/Max
I went with a much smaller box with a lower height.
I often scraped my leg hiking it over the old box on the screws holding the lid down.
It has 2 closures, but the added screws stops the rattling.
bicycles.stackexchange.com/users/36461/fixit7
I've always had the best luck using a milk crate.
I attach them to the rack using hose clamps.
Hose clamps won't break.
If I need something to be waterproof, I just put it in a plastic bag.
bicycles.stackexchange.com/users/45249/toddg
The "white container box" lasted about 2 months until it cracked in multiple places.  It was a cheap box(Dollar General variety).  New container is made by Sterilite and much sturdier.  I believe the old container cracked because of vibration and bumps transmitted from the ground to the box.  Can someone help me with ideas on how to cushion the new box?
Improve survivability of bicycle container
bicycles.stackexchange.com
<cargo><panniers>
bicycles.stackexchange.com/questions/63548
bicycles.stackexchange.com/users/36461/fixit7
Metal mesh baskets and wood are recommended for bike boxes with the addition of a plastic bag for waterproofing. Cheap containers do not last long however, a piece of rubber between the container and whatever holds it will damp some of the vibrations. The plastic boxes shown are thin and will break easily. You could dampen the vibrations with a piece of rubber under the box, use a smaller box or use thicker plastic (e.g. a milk crate). A metal wire basket is a sturdier option, and anything that needs to be waterproof can be put in a plastic bag. Wooden containers are also a possibility.
A metal wire basket is a sturdier option, and anything that needs to be waterproof can be put in a plastic bag. The plastic boxes shown are thin and will break easily. You could dampen the vibrations with a piece of rubber under the box, use a smaller box or use thicker plastic (e.g. a milk crate). Wooden containers are also a possibility.