Trains in France (e.g. Basel-Strasbourg) do run on strike days, though on a limited timetable. However on a strike day it is likely to be easier and cheaper to get from Interlaken to Strasbourg by train avoiding France and the strike completely. Look at https://www.bahn.com/en/view/index.shtml for trains from Interlaken to Kehl. From Kehl, frequent trams cross the new tram bridge over the Rhine into Strasbourg city centre.
Renting a car in Switzerland and dropping it in France is likely to incur a significant one-way drop-off fee, as the rental company will need to cover the cost of returning the car to Switzerland so it can be rented again.