The hardenable putty you're probably thinking of is a repair epoxy. You
can buy it by the stick or roll. It comes with an inner core and an
outer jacket, and when the two parts are mixed together, such as when
you knead it in your fingers, it gets sticky, and then it cures to form
a very hard waterproof repair patch.
The tape you're probably thinking of is electrical tape. People have
repaired pinhole leaks in pipes by wrapping the pipe with electrical
tape. As you say, this is a low pressure situation, so it doesn't take
much to stop a leak.
However, if the trap under the sink is plastic, then fixing the leak
won't involve anything more than replacing the "tail piece" which is the
vertical part between the sink and the trap, and might only require
tighting the trap adapter nut at the top of the trap. So, try
tightening the connections first.
If that doesn't do any good, then the correct way of fixing this
(depending on where it's leaking from) is usually to replace the tail
piece. In my experience, Waltec tail pieces are the easiest to install
if you use some silicone caulk as a lubricant for getting them in and
However, if this is a chrome plated brass drain, then you might as well
bite the bullet and have it all replaced. You see, that brass was paper
thin to begin with, and it's only the outside of the drain that's chrome
plated. The inside of the drain is bare brass, and brass is made of
copper and zinc, and zinc is a highly reactive metal. The chlorine in
city tap water gradually leeches the zinc out of the brass, and the
"metal" that's left is a reddish brown colour and very weak. If you
ever have the head of a bibb screw crumble under the force of a screw
driver when you're wanting to replace a rubber washer on a faucet,
you've come face-to-face with dezincified brass. So, if you patch one
leak, it won't be long before it starts leaking somewhere else.
Try the epoxy if you want cuz it won't cost much to try, and it might
work. But if this is chrome plated brass under your sink, the sooner
you replace it all with plastic, the better off you'll be.
You may have to cut a hole in the plaster or drywall under your sink to
connect to the drain pipe in the wall. In my building, they used a 1
1/4 inch threaded elbow in the wall, and soldered a threaded collar onto
the outside of the chrome plated trap arm. So, the threaded collar on
the trap arm screws into the threaded elbow on the 1 1/4 inch copper
drain pipe inside the wall. The problem is that the piece of brass
tubing sticking out of that threaded collar is way too weak to twist the
threaded collar out of the elbow. So, it often becomes necessary to cut
a fairly big hole in the wall under the sink so that you can unsolder
the threaded elbow in the wall, put it in a vice, take the threaded
collar out of it with a pipe wrench, clean up the threads with a rented
1 1/4 inch NPT tap, and then solder that threaded elbow back onto the 1
1/4 inch copper drain pipe. So, it's a big job, but if that's a chrome
plated brass p-trap and drain piping, it's a problem that's only gonna
get worse and is going to eventually need to be replaced anyhow.