Molecular line observations may serve as diagnostics of the degree to which the number density of cosmic ray protons, having energies of 10s to 100s of MeVs each, is enhanced in starburst galaxies and galaxies with active nuclei. Results, obtained with the UCL_PDR code, for the fractional abundances of molecules as functions of the cosmic-ray induced ionisation rate, $zeta$, are presented. The aim is not to model any particular external galaxies. Rather, it is to identify characteristics of the dependencies of molecular abundances on $zeta$, in part to enable the development of suitable observational programmes for cosmic ray dominated regions (CRDRs) which will then stimulate detailed modelling. For a number density of hydrogen nuclei of of $10^4$ cm$^{-3}$, and high visual extinction, the fractional abundances of some species increase as $zeta$ increases to $10^{-14}$ s$^{-1}$, but for much higher values of $zeta$ the fractional abundances of all molecular species are significantly below their peak values. We show in particular that OH, H$_{2}$O, H$_{3}^{+}$, H$_{3}$O$^{+}$ and OH$^{+}$ attain large fractional abundances ($geqslant 10^{-8}$) for $zeta$ as large as $10^{-12}$ s$^{-1}$. HCO$^{+}$ is a poor tracer of CRDRs when $zeta > 10^{-13}$ s$^{-1}$. Sulphur-bearing species may be useful tracers of CRDRs gas in which $zeta sim 10^{-16}$ s$^{-1}$. Ammonia has a large fractional abundance for $zeta leqslant 10^{-16}$ s$^{-1}$ and nitrogen appears in CN-bearing species at significant levels as $zeta$ increases, even up to $sim 10^{-14}$ s$^{-1}$. In this paper, we also discuss our model predictions, comparing them to recent detections in both galactic and extragalactic sources. We show that they agree well, to a first approximation, with the observational constraints.
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