You are on the right track. A simple single speed (no gears) beach cruiser with coaster brakes ("pedal backward" brake) with a step-through frame (for being able to stand up over it) and a lowered seat to begin with. My gut feeling is that a heavier bike is easier to balance since more of the weight is "stationary" as opposed to a squirmy human body, and newton's law, (bodies in motion tend to stay in motion).

Depending on how hilly it is where you live this single speed cruiser may be tough to pedal up hills, or not very fast, but that is not its purpose; its purpose is to build confidence.

This bike will be easier to become confident with, after which you will discover what kind of riding you prefer, and that will determine what kind of bike you will desire next. (Like an upright city bike, or a more speed oriented road bike). Your next bike will most likely have gears, which will almost always dictate hand brakes. (Although some internal-geared hubs have coaster brakes, but don't worry about any of that now.)

If it turns out you really do have some sort of physiological balance issues There are several types of adult tricycles from upright cruiser-style trikes with two wheels in back, to sportier, lower trikes with two wheels in the front.

But chances are your lack of "perfect" balance is a confidence issue, and a simple beach cruiser like you mentioned is a great bike to build your confidence.

I also liked the video boondocksaints posted.

Good luck and have fun.