Full or partial trisomy of human chromosome 21 results in dysregulation of gene expression, leading to the manifestation of specific phenotypes described in individuals with Down syndrome (DS). Defects in brain development, coupled with impairment in neurogenesis, are ultimately expressed as cognitive deficiency, Alzheimer disease (AD), and dementia. Amid the triplication of all human chromosome 21 (HSA21) genes, dual-specificity tyrosine phosphorylation-regulated kinase 1A (DYRK1A)-mediated neurogenesis and dendritic development have been attributed to the learning and memory deficits and cognitive impairment in the DS population. Upregulated DYRK1A perturbs the development and function of the brain, collectively affecting neurogenesis, synaptogenesis, synaptic transmission, and cell signaling pathways, which might disproportionately produce inhibitory neurotransmission and contribute to the cognitive phenotype. However, the lack of distinct gene-phenotype associations acts as a potential barrier to therapeutic improvement of cognitive performance and amelioration of AD-related neurodegeneration. The present review aims to summarize the neurogenetic consequences of triplicated DYRK1A in the DS population in relation to sexual dimorphism and expression of the Apolipoprotein Eε4 (APOE ε4) genotype. Notably, normalization of trisomic DYRK1A demonstrated improved synaptic plasticity, glutamatergic/GABAergic (excitatory/inhibitory) balance, and learning and memory in DS mouse models. Therapeutic approaches using inhibitors of DYRK1A, including catechins present in green tea extract and several other natural and synthetic agents, produced variable outcomes in cognitive improvement, depending on age and dose of administration. Mitigation of impairment in neurogenetic differentiation and cognitive performance might help control AD-related dementia and enhance quality of life. This review highlights the consequences of upregulated DYRK1A kinase on impairment of neurogenesis and cognitive deficits, and the therapeutic challenges associated with DYRK1A inhibitors for ameliorating dysregulated gene expression in DS models and human DS.