It cost ~4-5x times more than regular (today, not sure decades ago) but total cost of bridge build for it's majority was probably (overpriced) labor anyways.
In machinery stainless steels have universally poorer mechanical properties vs non-stainless alloys - a reduced corrosion rate is about the only benefit, and even so, it's still just "stain-less" — not "stain-free" — over the timescales of civil infrastructure, it would likely suffer damaging corrosion at some point.
The price ratio is roughly 6 historically, it has lowered to as low as 4 only recently, and stainless steel has been largely used, since the '80's, instead of conventional rebar steel.
But there is no stainless steel capable of having the same resistance/elasticity of cable steel.
To give you some comparative data in EU the normal rebar since more than 40 years is usually grade 44 or 45 that means that it starts elongating (and eventually fail) around 4300-4500 Kg/cm2 and breaks over 5400 Kg/cm2 (still as a reference normally the calculation uses a max of 2600 Kg/cm2).
Cable steel (again since more than 40 years), typically in 7 wire strands, is almost 4 times stronger, the same values are 16700 kg/cm2 and 18600 kg/cm2.
Typically it is pre-stressed to 12500-13600 kg/cm2.
Stainless steel cable is used in the fixed rigging (mast stays) of yachts precisely because of it's anti-rusting properties. Unfortunately the stainless still has to be completely replaced every 5-10 years because it weakens to the point you can't insure the yacht anymore.
Stainless has a very different failure mode to normal steel where a tiny nick in the surface of the stainless under tension forms a crack and corrosion occurs in the very bottom of the crack making it propagate deeper into the stainless until it fails while still looking shiny on the outside.
It cost ~4-5x times more than regular (today, not sure decades ago) but total cost of bridge build for it's majority was probably (overpriced) labor anyways.