I've cruised from Southampton twice and used WestQuay Parking, its part of a shopping center and is very secure. I drive quite an expensive car and never had any issues parking here. It costs around £6 per day , and an uber to the terminal was £4 and a normal taxi back to the car park was £7.
I paid £23 in total for 2 nights parking vs £75 for 2 nights parking at Southampton Terminal.
I would highly recommend parking here based on it being safe and very affordable.