This is a common problem in bathrooms that are rarely used. Bathroom plumbing systems are fitted with P-traps. These traps hold water that acts as a seal to keep out sewer gas odors. If a bathroom is not used for a prolonged time, the water in the P-trap can evaporate. Without the water seal, the foul odors can enter your basement and home. In most cases, you can solve the problem by letting the water run or pouring water down each drain. The water will block out the unwanted odors.
If your basement bathroom has a floor drain, you should remove the grate and make sure that the plug is in place. If the plug has been removed, which sometimes happens when sewer lines are cleaned, the sewer gases have a direct path into your home. You can purchase a replacement plug at any hardware store.
If these simple remedies do not solve the problem, you should make sure that you do not have a leak around the wax ring at the base of the toilet or a cracked or broken sewer line.