A 1GW reactor (nuclear fission reactors are usually 1GW, and you'll have a couple or several at a single plant) would produce something like 5-10 metric tons of gold in a year. According to USGS[1] there has been 187,000 metric tons of gold produced and another 57,000 metric tons known about but not yet mined.
There are currently hundreds of nuclear fission reactors in the world so if you assume a similar number of fusion reactors, you'll get thousands of metric tons per year of gold production. World gold production was ~3,600 metric tons in 2022 [2].
So ultimately, yes, it would impact the price, but it will be many decades before we have hundreds of fusion power plants producing all that gold, and so it's not going to saturate the market for a long time. It may never saturate the market if demand for gold continues to go up in the coming decades.
There are currently hundreds of nuclear fission reactors in the world so if you assume a similar number of fusion reactors, you'll get thousands of metric tons per year of gold production. World gold production was ~3,600 metric tons in 2022 [2].
So ultimately, yes, it would impact the price, but it will be many decades before we have hundreds of fusion power plants producing all that gold, and so it's not going to saturate the market for a long time. It may never saturate the market if demand for gold continues to go up in the coming decades.
1. https://www.usgs.gov/faqs/how-much-gold-has-been-found-world 2. https://en.wikipedia.org/wiki/Gold_mining#Statistics