Chilling stress in spring and mid-season heat stress are important environmental stresses that can significantly affect plant productivity. The objectives of this study were to understand the effects of cold (4 and 10 °C) or heat (30 and 40 °C) stress on biochemical and physiological traits in leaves and roots of basil (Ocimum basilicum L. cv. ‘Genovese’) young plants. After short-time exposure to mild and severe temperature stresses, both photosynthetic pigments’ and protein, as well as enzymatic and non-enzymatic defense components in basil leaves and roots, were quantified and compared with the control non-stressed plants. It was shown that both cold and heat treatment increase the content of chlorophyll a, chlorophyll b, and carotenoids. Chilling correlated with higher content of soluble proteins in leaves, whereas the concentration of these osmoprotectants in roots was higher under both cold and heat stress. For all tested antioxidant enzymes, higher activity was measured in leaves, and activity was related to temperature stress. SOD, CAT, A-POX, and P-POX activities was induced under heat stress, while the higher activity of SOD, CAT, P-POX, and G-POX was recorded under cold stress, compared to the control. In addition to the induced activity of enzymatic components, the content of secondary metabolites including total phenolics, flavonoids, and total anthocyanins, was several times higher compared to the non-stressed plants. Furthermore, total phenolic content was higher in roots than in leaves. Significant positive correlation can be seen among photosynthetic pigments, SOD, total phenolics, and flavonoids under severe temperature stress (4 or 40 °C) in basil leaves, while for roots, positive correlation was found in the content of secondary metabolites and activity of CAT or peroxidases. Obtained results are discussed in terms of phenotyping of O. basilicum cv. ‘Genovese’ response to heat and chilling stress, which should contribute to a better understanding of merged responses to cold and heat tolerance of this valuable crop.