Sink drain popup leaks drip due to the threads of the lower half of the popup assembly making a channel allowing the water to drain out.
The bathroom sink drain leaks into the vanity. The leak is coming from the nut where the drain attaches to the bottom of the bowl, so you tighten the gasket to the bottom of the bowl. Even though a tapered rubber washer should seal around the threads sometimes water travels through the threads and leaks out.
Plumbers putty seals the chrome trim to the inside of a porcelain or china sink. Marble or composite sinks use a thin foam gasket or silicone to make the same seal because putty can stain the more porous material.
The seal on the chrome trim in the bowl is important for filling the sink with water. However, it does not keep water from leaking below the sink.
The lower seal is key to a dry bathroom and can be difficult to achieve.
The lower seal is made with a nut and a thick tapered rubber washer at the bottom of the sink. A common leak, including yours, occurs when water uses the threads to sneak under the thick tapered washer. It usually is not much water, only a small drip, drip, drip.
To prevent this from happening, pack plumbers putty on the threads underneath the thick rubber washer. Loosen the nut, slide the rubber washer down, and pack plumbers putty on the threads between the washer and the bottom of the sink. When you tighten the nut, the putty will fill the void between the threads and the rubber washer.