In this study, methanol and water–methanol extracts of borage (Borago officinalis) herb dried using various methods were analysed for their phenolic profile and biological activity. Twelve compounds, including flavonoids (astragalin, kaempferol 4-glucoside, rutoside, and vitexin) and phenolic acids (caffeic, chlorogenic, 3,4-dihydroxyphenylacetic, ferulic, p-hydroxybenzoic, protocatechuic, rosmarinic, and syringic), were determined qualitatively and quantitatively in B. officinalis extracts by the HPLC-DAD method. The highest total flavonoid content was confirmed for the methanol extract from the hot-air-dried herb, while the methanol extract from the air-dried herb was most abundant in phenolic acids. The results of in vitro tests on human keratinocytes (HaCaT) and fibroblasts (BJ) showed that the extracts were able to reduce the intracellular level of reactive oxygen species in skin cells. Tests performed to assess inhibition of protein denaturation, lipoxygenase activity, and proteinase activity demonstrated that borage extracts have anti-inflammatory properties. In addition, the methanol extract of the herb dried in a convection oven showed the strongest inhibition of both collagenase and elastase activity, which is indicative of anti-ageing properties. The results show that the borage extracts are a source of valuable bioactive compounds with beneficial properties in the context of skin cell protection.