Do you hear water bubbling when it goes down (test by turning on water upstairs full force and allowing it to drain)? If so it’s probably partially clogged. For the smell itself you can pour bleach down said basement drain. Clogs usually get worse, so try to convince a plumber/sewer clearing company to come down as soon as possible. Or maybe a tree or something somehow got into the sewer line outside and is restricting flow, or even moved the pipe so that the pitch is not conducive to proper draining. Because even if you had back-flow from the storm, I don’t think the smell would persist, it would have flushed itself out by now.
I am not a plumber, just trying to help, so don’t rely on my “expertise”. Good luck!