Stainless is much more chemically resistant, so less likely to deteriorate from attack of various water-borne compounds, as well as common rust.
Stainless is much harder, so will minimize any wear from lip seals, set screws to hold bearings or face seals plus has less deflection which reduces bearing wear.
Stainless has a lower coefficient of expansion, so easier to work with when using press-fit bearings or seals.
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NO MATTER HOW WELL YOU HAVE DONE SO FAR, ALWAYS TRY TO BE BETTER TOMORROW.
Assume this is a 5-20 hp installation with a close coupled motor/pump with mechanical shaft seal. Smaller pumps under 5HP are usually built with plastic impellers and small hardware and stainless steel shafts as a standard in the purpose designed electric motor.
If you want to go cheap the carbon steel shaft will work. A little rust on the area of the shaft between the pump rotating mechanical seal and the shaft slinger that keeps moisture out of the motor bearing shouldn't hurt maintenance or shaft life. And often this area gets a coat of paint during the pump assembly process.
A high quality approach would call for bronze, 316 or even 304 stainless steel fittings such as impeller, impeller nut and washer as well as a shaft sleeve. These fittings should allow the avoidance of the expense of a stainless shaft if such material is not readily available from the motor manufacturer in the larger integral motor sizes 10hp and up.
If this chilled water pump is going into a high rise office building featuring high quality architecture and construction materials and a pump size over 25hp there is a chance the motors will not be close coupled and also that the pump will be specified as a horizontal split case type for maximum hydraulic efficiency. These pumps are usually bronze fitted (impeller, sleeves, etc. with a 1040 or 1045 steel shaft and a cast iron casing). Specifying stainless steel for the pump shaft will likely add unnecessarily to the pump cost and lead time.