What is the danger of small strands of stainless steel coming off the wire filter and being ingested?
Short answer: your probability of injury is less than 0.0002%
Long answer: per the estimates in this answer, approximately 360,000 Americans use a French Press. In the last few years, zero consumers have reported injuries due to drinking metal fragments from their press. There have been several reports of messes and even injuries due to broken glass carafes, but zero for anything to do with the wire mesh.
So whatever the incidence is of punctured internals due to French press use, it's clearly less than 1 in 360,000. We'll only be able to assign it a specific number when it actually happens. You are more likely -- at least 30 times more likely -- to be struck by lightning.
Of course, if you are using an old, second-hand knockoff carafe with a visibly damaged mesh, your personal danger is higher than that. It's also easily solved, buy a new mesh.
In the meantime, be careful with that carafe!