A lot of the old 42 Brunos had 1 1/2 shafts,and had more than 300 hp..Aquamet 22 is pretty strong.Some has to depend on how big of wheel you have and how much reduction.If its Aquamet, a 1 3/4 would be fine,but 1 1/2 could also work.
Reduction gears break shafts not so much HP. The rule of thumb is over 22: pitch on a single screw DE needs 1.75 shafting.
a 300 HP motor with a 1.5 shaft will most likely work as the wheel will be 22 or less. same HP with a 1.75 or 2:0 the load goes up to over 23" you will need a 1.75 shaft.
Engine torque and RPM in addition to what Capt Larry said. All of our yanmar 6LP's with a 2-1 were recommended to use a 1.5" shaft (Aquamet 22 without some crazy shaft length). Some Bertram 31 owners that had a 1 3/8" shaft used a 1.5-1 reduction. I never heard of one breaking. The same engine with a 2.5-1 reduction jumped up to a 1.75" shaft.