To add to Doremus' point. I have used old age-fogged paper under safe safelight conditions and while the development was taking place under safelight conditions it was not all all obvious that the paper was fogged. Not until I was more than half way through the fixer stage and had turned on the room-light, safely, was it clear that the paper was age-fogged.
A water wash may not stop development instantly so the paper continues to develop for a while in the water but water cannot fog paper, nor unfortunately can an acid stop bath prevent age-fogging either.
If you cannot be sure your safelight is safe then here's what to try. In total darkness, expose normally(i.e. under an enlarger light) then simply fix the piece of paper then put the light on to get your bearings if you need to. Leave paper to one side. Then again in total darkness expose under enlarger, develop, water wash and fix another piece of paper. Turn on the room-light and compare. The only fixed piece should be white and the developed, water stopped and fixed piece will be a shade of grey in the border area if there is age-fogging. If there is no signs of fogging then your safelight that you haven't used for this experiment but had used when processing, isn't safe. If there is greyness then this is age-fogging and how grey will depend on the extent of the age-fogging. Badly age-fogged paper will be quite grey and may lack contrast as well, making the print look flat and a bit insipid.
If the borders are only a very light grey and you appear to have good blacks and white, with a range of correct greys in the print such that contrast is still reasonable then get some benzotriazole and add to the developer. This is a rescue plan only and the print compared to fresh paper may not be as good but might be good enough.
pentaxuser