Re: 1981 Century Riveara,, Leak
If it's the same Riv model I used to drool over as a teen (in the eighties...I just loved those boats!) it's an I/O, and probably a Merc.
Assuming you've ruled out a hole in your hull or engine leak (does the boat leak only when running or anytime it's in the water?), that means you've got some checking to do where the outdrive mounts to the transom. There are large and small corrugated rubber tubes (bellows) which contain the drive shaft and shift cable, exhaust, as well as the smooth-walled cooling water hoses that pass through the transom.
All of these rubber parts degenerate over time and crack with age, movement, UV exposure, even animals like otters and beavers sometimes can chew on them.
Anyway, an 80s vintage boat will have been ready many times over to have all these parts changed out. Hopefully someone has kept up with some maintenance but it's obvious something is going a bit wrong.
You will have to contort your body pretty good to get your head down in that area on the inside to look for leaks coming in through the transom from any of those areas. It's possible the mounting hardware itself is leaking or you have leaks in several areas.
Not a surefire indication but external examination can give you some clue as to the condition of these parts. While on the trailer, raise the outdrive all the way and crank the wheel hard over. Get under the outdrive and look up inside where it connects to the hull. You should be able to see both bellows and the cooling lines. If anything looks aged or cracked, change it out or get it changed. This will require removing the outdrive and it can be a bit of a pain. However, we're talking a life and death issue for your boat until you get this sorted.
Here's a pic of an outdrive with one of the bellows visible inside the red box.
Here's a pic of a cracked shift cable bellows. That little tear right there will sink your boat at its mooring when the bilge pump finally overcomes your battery.
If you are getting it done professionally, have them check the drive shaft oil seals, gimbal torque and alignment.
Good luck, report back....and welcome to iboats!
You know by now that boat stands for Break Out Another Thousand, right...?
*Edit - seriously, though. If you are new to boating you'll want to get a handle on exactly the kind of maintenance your new baby requires - and the outdrive is a potential source of very expensive failure if not maintained. Find yourself a good marine mechanic (easier said than done sometimes) or, if you are a diy type, get the necessary engine and outdrive manuals and start reading. Also, head on over to the section of this forum specific to your power and outdrive package and look through some of the discussions to get a sense of what kinds of issues generally preoccupy owners of your type of engine/drivetrain.
Also, a boating safety or handling course is highly recommended for just general knowledge and competence.
Again, welcome!