When someone takes a shower in the upstairs bathroom, my sink in the downstairs bathroom fills and overflows with water. What would be the answer to correcting this problem?

There would appear, if this was not a problem previously, that they are on the same drain line. And at some point, there is partial blockage. The line needs to be cleaned. This would be done, depending on the plumbing layout, perhaps though the vent on the roof or other access.

I agree with Fr Chuck, you have a partial blockage in the drain line below the sink in the downstairs bath. Snake the line. I would remove the trap underneath the sink and snake from there.

Agree with both, Harold is correct in saying remove the trap to do your rodding.