You would attach the string to the balloon using parenting or you would put both the string and the balloon in the shape layer. I probably wouldn't use Puppet Pin to animate the string. I'll explain why later. I wouldn't use a rectangle for the string, I would use the pen tool to draw an open stroked path.
Once you have the stroked path selected you can add a shape animator by clicking the little Add button and add a Wiggle to the path you use for the string.
If the Balloon and the String are separate layers just parent them. If they are both on the same layer then you need to make sure that the Wiggle modifier is only applied to the string. Then just animate the position of the string.
To learn more about animating shapes type Shape layers in the Search Help field at the top right corner of AE and check out the community resources. You will find a lot better info that searching YouTube for help.
The reason that I would not use puppet pin as my first choice is that you cannot use any of the transform controls on a layer with puppet pin applied and the mesh would have to be quite dense to animate a string. If you did want to use puppet pin to create some organic movement of the string by recording your movements you would then have to pre-compose the puppet pin layer moving all attributes to the new layer then parent the nested puppet pin layer to the Balloon and animate the position of the parent.
One more thing. If you want to put some rotation animation in the balloon it would be a good idea to move the Anchor Point of the Balloon layer to the bottom of the artwork or you could use a null as the parent for the balloon and the string and position the null's anchor point where you wanted the center of rotation.
This seems like it will work perfectly! Very informative. I feel like I was certainly overthinking it in my head. I will put this to good use and give it a go, fingers crossed!