Research on the impact of neurocognitive disorders, such as Alzheimer’s disease, on rhythmic abilities, has yielded mixed results. This study employed a sensorimotor synchronisation task to examine whether neurocognitive disorder is associated with a specific deficit in beat induction, defined as actively inferring an underlying beat in music, as opposed to more automatic beat maintenance. Additionally, it sought to evaluate beat perception without production in this population. Eighty-seven patients (mean age = 81) from a geriatric day hospital underwent an evaluation of beat perception and a sensorimotor synchronisation task with three levels of beat saliency (BS; metronome; music with high BS; and music with low BS). We found that lower scores on the mini-mental state examination (MMSE) predicted a deficit in beat perception, as evidenced by an increase of false alarms in the Beat Alignment Test (p < .05). As expected, the highest synchronisation consistency (i.e., stability of the relative phase between taps and stimulus beats) was observed when tapping to a metronome, followed by high-BS music, and low-BS music (p < .001). Moreover, consistency decreased with lower MMSE scores, an effect found only with low-BS music (p < .001). Our findings suggest that beat maintenance, the dominant process when synchronising with a very salient beat, remains preserved in individuals with impaired cognitive functioning. However, older adults with neurocognitive impairment may experience difficulty with the perception of beat alignment as well as with beat induction, where both are necessary for extracting a less clear beat during synchronisation to music.