This putty is starting to crack and leak onto my basement ceiling. Is this repair as simple as removing old putty and applying new? Or is this practice now (or always has been) frowned upon and is there a better, more professional method of resolving this leak?
Plumber's putty would be used to attach the drain cover to the sink. It creates a blockage and suctions both pieces together.
It should be a ring of plumber's putty, when smashed is maybe a 1/2" wide.
What you have is a DIY disaster and looks like someone was trying to use plumber's putty to repair a possible leak. The problem is if it is exposed to "air" too long it eventually gets hard, dries out, and cracks... So not so good at repairing leaks in pipes.
I am guessing there is a leak in the pipe. You probably need to reset the drain cover with new plumber's putty. Once installed you should barely be able to see it from below. You will probably also need to fix the pipe leak which may mean buying a few pieces of PVC, primer/glue, and an adapter. If you are OK with doing it yourself it is a $15 job if that.
Plumber's Epoxy is what they should've used if they were going to do that.
Alternatively (providing there's space for it), use a pipe Repair Clamp.
But in your setup, it looks like there's enough couplings to unscrew, to make getting new parts in there comparatively easy. Which is what I'd recommend, instead of just adding more shenanigans.