Alginate-based electrospun nanofibers prepared via electrospinning technique represent a class of materials with promising applications in the biomedical and pharmaceutical industries. However, to date, the effect of alginate molecular mass and block composition on the biological response of such systems remains to some extent unclear. As such, in the present work, three alginates (i.e., M.pyr, L.hyp, A.nod) with different molecular features are employed to prepare nanofibers whose ability to promote cell adhesion is explored by using both skin and bone cell lines. Initially, a preliminary investigation of the raw materials is carried out via rheological and zeta-potential measurements to determine the different grade of polyelectrolyte behaviour of the alginate samples. Specifically, both the molecular mass and block composition are found to be important factors affecting the alginate response, with long chains and a predominance of guluronic moieties leading to a marked polyelectrolyte nature (i.e., lower dependence of the solution viscosity upon the polymer concentration). Subsequently, physically crosslinked alginate nanofibrous mats are first morphologically characterized via both scanning electron and atomic force microscopy, which show a homogenous and defect-free structure, and their biological response is then evaluated. Noticeably, fibroblast and keratinocyte cell lines do not show significant differences in terms of cell adhesion on the three mats (i.e., 30–40% and 10–20% with respect to the seeded cells, respectively), with the formers presenting a greater affinity toward the alginate-based nanofibers. Conversely, both the investigated osteoblast cells are characterized by a distinct behaviour depending on the alginate type. Specifically, polysaccharide samples with an evident polyelectrolyte nature are found to better promote cell viability (i.e., cell adhesion in the range 15–36% with respect to seeded cells) compared to the ones displaying a nearly neutral behaviour (i.e., cell adhesion in the range 5–25% with respect to seeded cells). Therefore, the obtained results, despite being preliminary, suggest that the alginate type (i.e., molecular structure properties) may play a topical role in conditioning the efficiency of healing patches for bone reparation, but it has a negligible effect in the case of skin regeneration.