I was there during a rail strike a few years ago. We were staying at the Hotel du Champs de Mars and were told that we had better plan to stay extra days, no way out of the city, etc...
But, there were plenty of taxis available early in the morning and on the way to the train station, we had a nice conversation in French where our driver told me their business was very slow during rail strikes because no one had anywhere to go. So, definitely set out early and you may have to walk to a taxi stand area. That's what we found. We had originally intended to walk all the way to the station if necessary.
Friends who lived in Paris told me I should go early and try to catch a TGV to Geneva, as was my plan that day, because some of the TGVs usually went even during a strike. They were right. Just as we walked into the Gare de Lyon, one was about to leave. We hopped on. When the conductor came by, we showed him our tickets for a later train and he punched them and we were fine. There were maybe a dozen people on that whole train so it was a surreal experience, but we got to Geneva even earlier than we had planned.
