Maintaining air supply is fundamental to staying alive, so how do we make sure of that? Obviously to minimise the risk of strangulation, avoid placing a rope over the front of the neck, or around the neck in a way that could apply pressure. Nooses are definitely out. The only safe configuration is a halter-neck style.
You should also be aware of ropes which may slip and end up around the neck. The risks are multiplied in suspension or if your partner falls or faints. Another obvious hazard is that a bound person can't remove a gag if it becomes a problem, and of course prevention is better than a cure so don't use anything that could be swallowed, become lodged in the throat or obstruct the airways. Their safety is your responsibility, so you need to be observant, as this kind of thing can turn bad very quickly. And that's a very good reason not to leave your partner unattended.
Breathing difficulties can be caused by constriction, for example of the ribcage; don't tie too tight. Remember that as you add wraps of rope, cinch bindings, or apply a load, tightness can increase. You should take extra care not to constrict the chest of somebody wearing a corset as they have nowhere left to breathe from.
In particular, suspension needs to be treated with extreme care, as it can turn normally comfortable ties into crushing constrictions. Suspension originated as a method of torture, it can result in serious nerve damage, or in extreme circumstances, kill in inexperienced hands. So don't try until you know exactly what you're doing - it's a whole different ballgame.
Certain positions can make breathing difficult. The degree of stress to which the person is subjected can also increase breathing rates. Never underestimate the effect being bound in a fixed position can have over time; it can stress the muscles used for breathing to the point of exhaustion. The condition can be exacerbated by stress, exhaustion, alcohol, drugs, and various conditions including obesity; this is called 'positional asphyxia', and is the major cause of custodial and recreational restraint deaths. Again it's easily prevented by observation. Most accidental deaths happen because the victim could not summon assistance.