A theoretical framework based on using the Frenet-Serret moving frame as the coordinate system to study the diffusion of bounded Brownian point-like particles has been recently developed [L. Dagdug et al., J. Chem. Phys. 145, 074105 (2016)]. Here, this formalism is extended to a variable cross section tube with a helix with constant torsion and curvature as a mid-curve. For the sake of clarity, we will divide this study into two parts: one for a helical tube with a constant cross section and another for a helical tube with a variable cross section. For helical tubes with a constant cross section, two regimes need to be considered for systematic calculations. On the one hand, in the limit when the curvature is smaller than the inverse of the helical tube radius R, the resulting coefficient is that obtained by Ogawa. On the other hand, we also considered the limit when torsion is small compared to R, and to the best of our knowledge, the expression thus obtained has not been previously reported in the literature. In the more general case of helical tubes with a variable cross section, we also had to limit ourselves to small variations of R. In this case, we obtained one of the main contributions of this work, which is an expression for the diffusivity dependent on R', torsion, and curvature that consistently reduces to the well-known expressions within the corresponding limits.