What dimensions does the base need to be for a standing post?
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&quot;x 3.5&quot;) with a 4x4 post (actual size 3.5&quot;x 3.5&quot;). Therefore that box is 19 inches high (including 1.5&quot; for the feet on each corner). The width based on the 4x4 post is about 16&quot; wide with an interior dimension of 13&quot;x13&quot;.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&quot; 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.