The quantity of mesenchymal stem/stromal cells (MSCs) required for a particular therapy demands their subsequent expansion through ex vivo culture. During in vitro multiplication, they undergo replicative senescence which may alter their genetic stability. Therefore, this study was aimed to analyze cellular, molecular, and chromosomal alterations in Wharton’s jelly-derived MSCs (WJ-MSCs) during their in vitro sequential passages, where WJ-MSCs were sequentially passaged up to P14 and cells were evaluated at an interval of P2, P6, P10, and P14. They were examined for their morphology, tumorigenicity, surface markers, stemness markers, DNA damage, chromosomal aberration, and telomere length. We have processed five full-term delivered human umbilical cord samples to obtain WJ-MSCs. Morphological appearance observed at initial stages was small fine spindle-shaped WJ-MSCs which were transformed to flat, long, and broader cells in later passages. The cell proliferation rate was gradually decreased after the 10th passage. WJ-MSCs have expressed stemness markers OCT-4 and NANOG, while they showed high expression of positive surface markers CD90 and CD105 and lower expression of CD34 and CD45. They were non-tumorigenic with slow cellular aging during subsequent passages. There was no chromosomal abnormality up to the 14th passage, while increase in comet score and decrease in telomere length were observed in later passages. Hence, our study suggests that early and middle passaged (less than P10) WJ-MSCs are good candidates for clinical administration for treatment.