Only for a short period of time maybe for 2-3 days after you are done.The stain fumes get moving around and come up the furnace vents,and it collects in the tumble part of the dryer.
Take out your furnace filter and lint trap will help,dont leave them in the basement.
When you are completely done within 2-3 days the fumes will be pretty much be gone.

You will also need to turn out the pilot light when you are staining in a confined area such as a basement. As the fumes from an oil based stain can build up and cause a fire hazard.