Leaking faucet connection (inside vanity)
First observation I see is that the basin is most likley not level as the the black drainpipe is not entering the trap flush. (it is cocked sideways and this is most likely because the basin isn't level). So to this note, is the cabinet firm and secure? Is anyone (small child or a teen) climbing up onto the countertop or basin? If so, put an end to it. If water is being allowed to sit on the back of the counter, it will over time rot the back portion of the wood which will weaken it, allowing the same thing to occur. You should clean out under the sink, get a bright flashlight and check for any water damage or areas where the counter is separated from the backsplash or wood supports. *(sometimes these are just wooden blocks glued on).

Once you've fixed or confirmed the stability and levelness of the basin, remove the sink drain and black pipe. Check the drain/pipe for any cracks, replace if necessary. Check the seal/quality of the black rubber/foam gasket, replace it if necessary, and clean all surfaces. Also, pickup some fresh plumbers putty at the local hardware store and apply fresh new putty graciously under the drain where the drain and portion that penetrates into the basin. This will help tremendously.

Reassemble everything, firmly, (tight but not overtightened), and clean off any excess. Make sure putty does not impede or protrude fromthe botton of the basin before you re-secure the washer and black nut. If you overtighten it, the drain, the nut, the tubing, may crack and you'll have to replace it.

Once reassembled, everything should be square, and you should be leak free. If not, inspect your work and redo it.