Cosmic rays are an essential ingredient in the evolution of the interstellar medium, as they dominate the ionisation of the dense molecular gas, where stars and planets form. However, since they are efficiently scattered by the galactic magnetic fields, many questions remain open, such as where exactly they are accelerated, what is their original energy spectrum, and how they propagate into molecular clouds. In this work we present new observations and discuss in detail a method that allows us to measure the cosmic ray ionisation rate towards the molecular clouds close to the W28 supernova remnant. To perform these measurements, we use CO, HCO+, and DCO+ millimetre line observations and compare them with the predictions of radiative transfer and chemical models away from thermodynamical equilibrium. The CO observations allow us to constrain the density, temperature, and column density towards each observed position, while the DCO+/HCO+ abundance ratios provide us with constraints on the electron fraction and, consequently, on the cosmic ray ionisation rate. Towards positions located close to the supernova remnant, we find cosmic ray ionisation rates much larger (≳100) than those in standard galactic clouds. Conversely, towards one position situated at a larger distance, we derive a standard cosmic ray ionisation rate. Overall, these observations support the hypothesis that the γ rays observed in the region have a hadronic origin. In addition, based on CR diffusion estimates, we find that the ionisation of the gas is likely due to 0.1−1 GeV cosmic rays. Finally, these observations are also in agreement with the global picture of cosmic ray diffusion, in which the low-energy tail of the cosmic ray population diffuses at smaller distances than the high-energy counterpart.