How do I seal the sink basin opening?
It should not be necessary to try to goop some type of sealer around the lower neck of the sink drain. Such solution may work but it is not the desirable approach when you consider that someday this assembly will need to come apart for repairs. 

In looking at the assembly that you have I wonder if one or more things are at play here.

1) If the drain pipe tail was not centered in the neck of the sink drain and then had lateral force on it due to the down stream goose neck trap it could be forcing the gasket open on one side. Check that there is no undue large amount of sideways pressure on the lower part of this pipe assembly.

2) It looks like the gasket directly contacts with the face of the capture nut around the drain pipe. It is possible that by tightening the capture nut that it is putting a twisting force on the gasket that may cause it to "fold away" from the sealing edge of the sink neck.

3) It is possible that there could be a small crack or pitted area on the sink neck that keeps the rubber gasket from engaging 100% with the sink neck.

4) Sometimes the "rubber" used in this type gasket is made of a plastic material that is anything but rubber. Check at the local hardware store to see if they have another gasket that is more pliable. In some cases a silicon material may be better.