Sorry to hear about your problem.
The only way any smells from the basment could enter the system, is if the ductwork is leaky, meaning that the connections are loose or even diconnected, in the basement. If the ductwork is relatively tight, without any large air gaps or disconnections, then I would be looking at the bathroom that is closest to the return air grille.
The return-air grille draws air from the surrounding space to circulate it thru the ductwork. When a bathroom is nearby it will draw air from there, as well.
If the sink/ tub/ shower is not used often, the water that fills the trap, to prevent sewer gases from coming up thru the drain, can slowly evaporate, allowing an air space for sewer gases to escape. These gases will be pulled towards the return-air grille, sucked thru the ductwork and will dump out at all of the registers. To solve this, run the sink, tub/ shower faucets for a minute, at least once a month, to keep the traps full.
Another possible culprit, is if the wax ring on the bottom of the toilet is not sealing well, the sewer gases can be pulled past the wax and towards the return grille. To solve this, involves removing toilet and replacing wax ring. This is a little more involved....so make sure all the traps are full of water first and test it.
If you need further help or clarification on this answer, please do not hesitate to reply to this post.