Should I put checked wire under a raised garden container?

I'm not sure with the mesh wire would be for? I have one in my yard and I did put all bagged garden soil in it, but you probably wouldn't have to. The reason I did it was to keep down the weeds

Put landscape fabric down first then fill with the garden soil making sure there is enough soil for your plants.

We did something similar although about 4 times larger. We cut up cardboard boxes for underneath, and it's been 3 years and haven't seen much weed sprouting at all. We did, however, use bagged garden soil (Miracle Grow) because it is sterilized of weed seeds, then nutrients added. We use miracle grow weekly on the plants too. No wild animals have burrowed under it (luck?) if that's what your worry is, and we have armadillos here. Go for it, use the best stuff you can, it will be worth it. Pic is one corner. We planted peppers in the brick holes.

I use large sheets of cardboard in the bottom of my raised beds for weed block. If you put hardware cloth in the bottom as you noted it will serve to stop any digging critters from getting to your plant roots. I fill my beds with a mixture of garden soil, composted manure and vermiculite to keep the soil from compacting too much. This is truly the expensive part of raised beds, but it's generally a one time expense. Hope this helps!