Flimsy Cacti? If the cutting was taken from an under watered cacti, there could be a loss of turgid pressure. The cutting would look sucked up and be a bit rubbery. It is more likely that the cutting is just thin for its length, like you said. Not too much can be done if this is the case.
For a calloused 12in typical cutting, its normal to bury the bottom 2-3in or so in soil. It can be more or less but the idea is to provide conditions for photosynthesis and a healthy root system. Stake the cutting or bury deeper if its liable to fall over once potted in dry soil. Roots will provide stability.
Once the cutting has rooted in partial sun, water and gradually expose it to more sun.
Girth is mostly dictated by age. If there is a thin spot, it will not likely grow much thicker. All you can do is provide water, nutrients and most importantly, sunlight.
"It is my quiet hope that the psychedelic drugs will give us that guidance towards the understanding of the mind. They just might let us see that trail through the dark forest where most of the people who search choose to follow the lighted path." --TiHKaL