I bought a rust converter in Brico a several years back ( can't remember the name and it's in Spain and sadly I am not ) to use on our swing seat and it actually worked very well, so worth a try. Not been out now for just on two years so don't know what it will be like now but hoping it will still be ok, if not I'll just do it again.
Powder coating would be a great option if done correctly but cannot be done in situ, you would have to take the staircase to a professional coater for treatment, will be a costly exercise but if done correctly will last for years and years.