We present a multiwavelength study of the molecular gas properties of a sample of local Seyfert 2 galaxies to assess if, and to what extent, the presence of an active galactic nucleus (AGN) can affect the interstellar medium (ISM) properties in a sample of 33 local Seyfert 2 galaxies. We compare the molecular gas content (M_H_2_) derived from new and archival low-J CO line measurements of a sample of AGN and a control sample of star-forming galaxies (SFGs). Both the AGN and the control sample are characterized in terms of host-galaxy properties, for example stellar and dust masses (M* and Mdust, respectively) and the star formation rate (SFR). We also investigate the effect of AGN activity on the emission of polycyclic aromatic hydrocarbon (PAH) molecules in the mid-infrared (MIR), a waveband where the dust-reprocessed emission from the obscured AGN contributes the most. The AGN hosted in less massive galaxies (i.e., M*<10^10.5^M_{sun}_; Mdust<10^7.5^M_{sun}_) show larger molecular gas contents with respect to SFGs that have the same stellar and dust masses. When comparing their depletion times (t_dep_{prop.to}M_H2_/SFR), AGN show t_dep_~0.3-1.0Gyr, similar to the times observed in the control sample of SFGs. Seyfert 2 galaxies show fainter PAH luminosity the larger the dominance of the nuclear activity in the MIR. We find no clear evidence for a systematic reduction in the molecular gas reservoir at galactic scales in Seyfert galaxies with respect to SFGs. This is in agreement with recent studies that show that molecular gas content is only reduced in sub-kiloparsec-sized regions, where emission from the accreting supermassive black hole dominates. Nonetheless, we show that the impact of AGN activity on the ISM is clearly visible as a suppression of the PAH luminosity.