Train from airport to Southampton central, free bus to town quay an red jet...
That would be my route.
Post what your rough timings will be, you never know, there's enough of us around this part of the South coast that one of us may be able to help you out.
If your at Southampton Airport the train station is right there about 2 min walk from the exit of the terminal building, so hop on a train to Southampton Central, then short 5 or 10 min walk to the red jet ferry terminal and a quick high speed journey across to Cowes.