The proper tool to interface with that bottom bracket is one like this:
The pictured tool is a Park HCW-18 and runs about $15 USD.
There should also be a large nut that threads on top of the bottom bracket piece (an adjustable bearing cup) in your picture. This nut is used to jam (tighten) against the adjustable cup and lock its adjustment in place. Without a jam nut, the adjustment you make to the bearing cup will not be maintained.
I was going to find an image of this jam nut on a one-piece crank, but I actually found something better: A link to a Park tool webpage that includes a video on how to do the adjustment you are seeking.
Bottom Bracket Service - One-Piece Crank
The way to adjust this bottom bracket is a little trial-and-error to get it just right as the tightening of the jam nut always imparts some change in the adjustment. When it is correct, the bottom bracket should have no play in it, but NOT be overtightened such that there is perceptible drag due to increased friction from the over tightened bearings. Also note that some of these one piece cranks/bottom brackets may not be of the highest quality and sometimes you cannot get a perfect adjustment in that you would either have to accept a tiny bit of over-tightened bearings or just slightly loose. Personally, when dealing with this situation, I would prefer slightly tight over loose bearings, and I am stressing "slightly" as in barely perceptible.