If $B$ denotes the binormal vector, $S$ denotes the arc length, $\tau$ denotes torsion, and $N$ denotes principal normal vector, then,
$$\frac{dB}{dS} = -\Bigg|\frac{dB}{dS}\Bigg|N = -\tau N$$
So if $\tau$ is the magnitude of $\frac{dB}{dS}$, i.e. a modulus value, then how can it be negative? It is written here that "if torsion is positive, the curve "turns" to the side to which binormal vector points. If torsion is negative, the curve "turns" to the opposite side."
The definition of the torsion of the question, i.e. $$\frac{dB}{dS} = -\Bigg|\frac{dB}{dS}\Bigg|N = -\tau N$$ is not correct.
What is correct is that
$$\tau = -\frac{dB}{dS} \cdot N.$$ With this correct definition, the torsion can be positive as well as negative. See wikipedia for additional details.