To hang string lights you shouldn't need a lot of weight in the bases unless you're stringing 100' or so. For an average size yard I think what you show in the picture should be fine. I assume that you'll fill the base with with a counter-weight of stone or sand. You could use them as planters but then you'll want to put a liner in to protect the wood from the soil and moisture.
I would do large river rock in case you want to move the posts into a new configuration. It'll be a lot easier to unload and reset than sand or soil.
The structure in the picture is made up of 2x4s (actual size 1.5"x 3.5") with a 4x4 post (actual size 3.5"x 3.5"). Therefore that box is 19 inches high (including 1.5" for the feet on each corner). The width based on the 4x4 post is about 16" wide with an interior dimension of 13"x13".This might be a little off but not by much.
As I said above I believe that filled with rock or sand that will be a sufficient base for the posts to support light strings of 25 or 30 feet (you didn't indicate length of light string). However, it wouldn't hurt to over-build the base to a 24" square box with the same height.
Other factors to consider are:
- The ground under the base needs to be level.
- The post needs to be securely supported within the base (can't see inside). The river rock I suggested will help support it.
Possibly a StackExchange engineer might want to weigh in on this.