Will this frame support hanging planters?
To add stability you could use 2x6 members for the cross pieces along the bottom. Then cover the bottom with a half of a sheet of plywood. Pour concrete into the box thus formed at the base by the 2x6's and plywood. 

You could get by without having to go all the way to full angle braces across the sides (which would get in the way of your hanging plants anyway) by installing some gussets made of thin plywood as shown below:



At a suitable size these would probably come right up to where the first hanging pot wires go.

I see that you have each corner made of just one 2x4. I would encourage you to make the corners out of two 2x4's attached in an L shape in each corner. This will go a long ways toward making the unit a good bit sturdier and will also make the cross bars on two of the sides have a whole lot more wood to bolt into.



I would echo that the proper fasteners for this structure are going to be 1/4" or 5/16" diameter carriage bolts as opposed to screws or nails.