High-pressure homogenization (HPH) is one of the food-processing methods being tested for use in food preservation as an alternative to pasteurization. The effects of the HPH process on food can vary depending on the process parameters used and product characteristics. The study aimed to investigate the effect of pressure, the number of passes, and the inlet temperature of HPH processing on the quality of cloudy blackcurrant juice as an example of food rich in bioactive compounds. For this purpose, the HPH treatment (pressure of 50, 150, and 220 MPa; one, three, and five passes; inlet temperature at 4 and 20 °C) and the pasteurization of the juice were performed. Titratable acidity, pH, turbidity, anthocyanin, vitamin C, and total phenolics content, as well as colour, and antioxidant activity were measured. Heat treatment significantly decreased the quality of the juice. For processing of the juice, the best were the combinations of the following: one pass, the inlet temperature of 4 °C, any of the used pressures (50, 150, and 220 MPa); and one pass, the inlet temperature of 20 °C, and the pressure of 150 MPa. Vitamin C and anthocyanin degradation have been reported during the HPH. The multiple passes of the juice through the machine were only beneficial in increasing the antioxidant capacity but negatively affected the colour stability.