Unidentified pipe and smelly water trickling under slab into basement floor drain
The 3/8” pipe is a “primer” for the sewer line. We use primers when we’re afraid the “P-trap” will dry-out and allow smells to infiltrate up through the sewer line and into the house. 

The 3/8” line is a little large, (we use 1/4” where I live) and it’s connected to a cold water line by a sink, toilet, etc. close to the drain. If you can determine where it’s connected, you could put a valve on it and slow the water flow...being careful not to turn it down too much and allow the drain to dry out. 

The smell could be from wet/damp soil around the drain and seeping into the house through cracks between the drain and concrete.