Locate the sink drain where it enters the wall behind the sink. This is the standard place to attach a bathroom vent. Installing a vent here will serve to prevent air locks for the entire basement bathroom.
Attach a 2-inch sanitary T-fitting in the stud space where the sink drain connects to the drain line. A sanitary T has three openings: one connects to the horizontal sink drain, one fits to the drain line below, and the other extends upwards for the vent pipe.
Install a 2-inch PVC pipe to the top of the sanitary T-fitting and run the pipe straight up between the wall studs until the pipe is between the joists above. Cut the pipe with a small handsaw.
Attach a 90-degree PVC corner fitting to the end of the pipe and turn the corner so the other end of the fitting faces the nearest exterior wall.
Cut a pipe 4 inches longer than the distance between the 90-degree corner fitting and the rim joist. The rim joist is the large board that sits on edge and provides a frame all the way around the floor joists. The rim joist serves as the barrier between the inside of the house and the outside.
Drill a hole in the rim joist with a power drill, fitted with a 2-inch hole bit. The hole must sit higher than the 90-degree fitting, so that the pipe slopes at the rate of 1/4 inch per lineal foot. For example, if your pipe is 4 feet long, the hole must be 1 inch higher than where it connects to the 90-degree fitting.
Run one end of the pipe you cut through the hole and connect the other end to the 90-degree corner fitting.
Attach another 90-degree corner fitting to the end of the pipe that extends about 2 inches out from the rim joist. The opening on this fitting should face straight down.
Measure and cut a 3-inch segment of PVC pipe and attach it to the corner fitting. The pipe will point toward the ground.
Attach a 2-inch screen cap to the end of the PVC pipe. The screen will prevent insects from entering the pipe.
Apply clear silicone caulking around the pipe where it exits the rim joist.