Aims. We examine whether dust mass can trace the total or molecular gas mass in late-type Virgo cluster galaxies, and how the environment affects the dust-to-gas ratio and the molecular fraction. Methods. Using the far-infrared emission, as observed by the Herschel Virgo Cluster Survey (HeViCS), and the integrated HI 21-cm and CO J = 1-0 line brightness, we infer the dust and total gas mass for a magnitude limited sample of 35 metal rich spiral galaxies. Environmental disturbances on each galaxy are considered by means of the HI deficiency parameter. Results. The CO flux correlates tightly and linearly with far-infrared fluxes observed by Herschel, especially with the emission at 160, 250 and 350 μm. Molecules in these galaxies are more closely related to cold dust rather than to dust heated by star formation or to optical/NIR brightness. We show that dust mass establishes a stronger correlation with the total gas mass than with the atomic or molecular component alone. The correlation is non-linear since lower mass galaxies have a lower dust-to-gas ratio. The dust-to-gas ratio increases as the HI deficiency increases, but in highly HI deficient galaxies it stays constant. Dust is in fact less affected than atomic gas by weak cluster interactions, which remove most of the HI gas from outer and high latitudes regions. Highly disturbed galaxies, in a dense cluster environment, can instead loose a considerable fraction of gas and dust from the inner regions of the disk keeping constant the dust-to-gas ratio. There is evidence that the molecular phase is also quenched. This quencing becomes evident by considering the molecular gas mass per unit stellar mass. Its amplitude, if confirmed by future studies, highlights that molecules are missing in Virgo HI deficient spirals, but to a somewhat lesser extent than dust. © 2012 ESO.