To expand on the other answer. There is not going to be any fix that you can somehow kludge on top of the already leaking basket that will provide a reliable solution. The correct solution is to simply take apart the existing assembly and repair it with a proper sealant.
Note that on most sinks there is to be some plumbers putty installed under the upper rim of the drain basket that gets squeezed down against the rim of the hole from the top side of the sink. This putty is what really seals the joint that is leaking on your sink. The washer that you see underneath the sink is just a bearing surface for the big round ring nut that threads onto the outside of the drain basket.
In many instances the drain basket may be best replaced completely. They are not particularly expensive and can be purchased at hardware stores and home centers.
Keep in mind that if you should choose to ignore the advice here and attempt to fix the leak by globbing on silicon sealer, epoxy or other type of material you will make it much harder to remove the basket for the proper fix when the kludged repair fails.