We present the submillimeter spectra from 450 to 1550 GHz of 11 nearby active galaxies observed with the SPIRE Fourier Transform Spectrometer (SPIRE/FTS) on board Herschel. We detect CO transitions from J(up) = 4 to 12, as well as the two [C I] fine structure lines at 492 and 809 GHz and the [NII] 1461 GHz line. We used radiative transfer models to analyze the observed CO spectral line energy distributions. The FTS CO data were complemented with ground-based observations of the low-J CO lines. We found that the warm molecular gas traced by the mid-J CO transitions has similar physical conditions (n(H2) similar to 10(3.2)-10(3.9) cm(-3) and Tkin similar to 300-800 K) in most of our galaxies. Furthermore, we found that this warm gas is likely producing the mid-IR rotational H-2 emission. We could not determine the specific heating mechanism of the warm gas, however, it is possibly related to the star formation activity in these galaxies. Our modeling of the [C I] emission suggests that it is produced in cold (T-kin \textless 30 K) and dense (n(H2) \textgreater 10(3) cm(-3)) molecular gas. Transitions of other molecules are often detected in our SPIRE/FTS spectra. The HF J = 1-0 transition at 1232 GHz is detected in absorption in UGC 05101 and in emission in NGC 7130. In the latter, near-infrared pumping, chemical pumping, or collisional excitation with electrons are plausible excitation mechanisms likely related to the active galactic nucleus of this galaxy. In some galaxies, few H2O emission lines are present. Additionally, three OH+ lines at 909, 971, and 1033 GHz are identified in NGC 7130.