Should I attempt to fix a sink leak myself?
Are you talking about the threaded joint between the white pipe and the gray pipe leading down from the sink?  You could just try snugging the joint up by turning the nut clockwise.  If it is loose you can do it with your hands and then just snug it up with a large pair of slip joint pliers or a pipe wrench.  Be careful not to over do it and break anything.  All those joints are like compression fittings.  The mating ends of the pvc pipes are compressed against each other by the plastic nut that you can see on all the joints.

If the joint is just seeping, chances are that just a little tightening will stop it. 

If you are talking about the black pipe that connects to the bottom of the sink you may have to loosen the nut connecting the black pipe to the gray one so that you can tighten the black pipe up against the sink.  There is a threaded pipe inside the sink (the metal pipe that the sink stopper goes in) that threads into that black pipe and holds it firmly to the sink.  Normally plumbers putty is placed as a sealant under the flange of that pipe, which is then inserted into the drain hole in the bottom of the sink and then threaded into the black drain pipe you can see in the photo.

Again, a little snugging up may be all that is needed to stop the seeping water.  If not you will have to disconnect the plumbing under the sink so you can remove the drain pipe from inside the sink and reapply plumber's putty and reinstall.  Look for Youtube videos about installing sinks for more details.