I went to replace a washer in the cold water bathroom basin faucet, and saw that the cup at the end of the stem was deteriorating. So, I put in a new stem which came with a new washer, and I also put in a new seat, but I still have to turn the faucet handle kind of tight to fully stop the spout drip. I don't know where to go from here.

it seems as if you did everything correctly. if it stops dripping then the place where the seat screws into the faucet body is tight and not letting water thru.
give it a couple of days and if it doesn't get better, change the washer.
not much else you can do..

Thanks for the reply, but just for grins, I just replaced the washer that came installed in the new stem with another new washer from my collection. It works fine now! Nothing is ever easy :-( I have a feeling that the old seat was OK, but since I bought a lifetime supply of seats, I'm not going to put the old one back to find out.

Washers are made from different types of Rubber. Some are really hard some aren't as hard.. The softer ones may wear out sooner, I don't know for sure but I think the harder ones are cheaper for the manufacturer, I am guessing because it seams that most manufacturers are in a race to the bottom with their quality.