The definition of DON´T is a command or entreaty not to do something
As you are seeking guarantees here, not an order it´s signaled by "not to do", it´s less overbearing/demanding. It let´s open to the possibility of "not doing" (promise not fall - you may fall or not).
It´s a bit similar to "or not
?" that leaves open to another possibility that is yes/yes sir!https://dictionary.cambridge.org/grammar/british-grammar/questions-and-negative-sentences/nothttps://www.easypacelearning.com/all-lessons/english-level-2/1341-promise-learning-how-to-make-and-promises-english-lesson
In this case, you can´t positively promise that you won´t fall/are not falling (anytime). Indeed, the person is asking you to "try not to fall". Again, there is another possibility open here, that of falling...