How do I get rid of mold and mildew smells that are in the basement?

Sniff out where it is coming from and try to figure out if you have water sleeping in somewhere or just from constant humidity. A dehumidifier would help keep the humidity down, Once you find where the odor is coming from and resolve the moisture issue, spray vinegar on the carpet, fabric on furniture, drapes, whatever surfaces are around. Make sure you check to make sure it won't change the color or texture in an inconspicuous spot. Febreze for fabric will help some, too.

Try to remove sourc of the problem in some circumstances a dehumidifier is needed after the clean up.

If you have those little basement windows put a fan in two on opposite sides of the basement to create a cross draft, one blowing into the room and one exhausting the air out. https://www.youtube.com/watch?v=YR-mkF-DvZQ You can also use glass block and a small clip on fan.

Then address the moisture problem that might include 1) sealing the concrete floor to prevent moisture wicking into the room, 2) repair any cracks in the basement walls and painting with Drylock to prevent water coming through walls, 3) add a dehumidfier.

If none of those work, you might have to seal the outside basement walls and add a French Drain around the house.... check Apple Drains videos on YouTube.