Fortunately, today, thanks to the miracle of things like RTV (room temperature vulcanizing) and self-sealing caulking, your job is lots easier than just 35 or 40 years ago.
Back in the day (as they say), keeping up a wooden craft involved not only ensuring that all of the seams were properly caulked (in this case we're talking about a different type of caukling material) but that the caulking was dressed, dried, primed and painted correctly.
At that time, if you found your craft had a leaky seam, you had to open up that seam and get some caulking material -- usually hemp or something similar -- and than taking a small chisel or paint knife and hammer and stuffing the strings into the leak. The caulking material was available in string form.
Once you finished that, you then had to close the seam with either a hammer and small brass or stainless steel track or screw and you would then, after making sure that the new tacking was properly countersunk, fill the indentation with a fiberglass or wood putty material that had to cure in a dry room at over 70 degrees. This usually took about 24 hourse.
Once this was finished, you then had to take either a small belt or orbital sander, being very carful to apply very light pressure on it, sand the area smooth and then you would have to prime the area with a good marine base paint, usually an oil-based primer.
You'd then have to let the primer set for about 24 hours before you could apply the one or two topcoats that were needed.
Then came the real test, hauling it back to the water to see if it continued to leak or you had patched up the leak.
Now came the real problem because, unless the seam was at the keel of the boat, you would have to sand, prime and paint the rest of the boat (actually not a bad idea, either).
Today things are a lot easier thanks to modern polymer-based caulking tools that let you run a bead of caulk into and along the offending seam and then, after proper curing and drying -- in some cases as quickly as four hours -- you can then sane prime and paint the seam with a good marine-grade primer which usually dries in an hour or so, if the temperature is right and, if you buy the right primer/topcoat combination, all you have to do is apply the topcoat paint and let it cure before you haul it back to the water, very secure in the knowledge it won't sink.
Imagine, though, that you had a lapstrake craft. The upkeep on those craft in the old days was humunous as each seam had to be caulked, nailed down, if it was needed, sanded, primed and painted. It was a huge investement in time.
They still have lapstrake boats out there, but, thanks to the same modern polymer-based caulking materials you have and the primer/topcoat teams of paint you can pick up at a marine supply store, you're on your way in no time.