See if they have a small piece of rubber you could cut and install. If not, any small tube of "Rubber Sealant" will work. Get clear.
Permanent? Is anything permanent?
How are you planning on getting any sealant where the drain meets the sink if you don't plan on taking it apart? My suggestion for your desire to do it right,
place a bucket under the drain to catch the leakage and dump regularly.
How are you planning on getting any sealant where the drain meets the sink if you don't plan on taking it apart? My suggestion for your desire to do it right,
Then what you are looking for is something like "Mighty Putty" or "Leak Ender 2000"? Don't know where these are sold other than on TV and I know not how they would work.
If the pipes are in such condition that you don't want to mess with them, don't touch anything, use a bucket for now, get all the parts you need, THEN take it apart and replace. Chances are if the pipes are at all dirty and/or wet, anything you use probably won't hold anyway. If you get a 2-part ribbon epoxy, that will adhere wet or dry, but you'll never get it off without more damage.
Why does it need to be fixed now if you aren't going to do it right? If the issue is that it won't hold water get a small rubber mat (or one of those things that help you open jars) and place over the drain before filling.