It's a purely gut feeling about what the PO tells you about the bike. I just bought an '84 with 23,000 miles and it looks like brand new. The PO tells me it was kept inside a heated walk-out basement and only ridden on fair days. The way the bike looks and runs I can surely believe what he told me.
I've seen these bikes go for $500 up to $5000, depending on condition, of course. I just paid $3500 for mine, and think I got a good deal. Mine is also loaded with lots of Markland accessories, and everything looks new.
So, the condition of the bike compared to what the owner tells you, plus how much you trust him anyway, should give you a feel for the sale.
If I may ask, why an '84 specifically? "83, '84 and '85's were all identical with no changes. "86 is when they came in the different colors scheme and red upholstery. '86 is also when they introduced the cruise (speed) control on them.
I googled "Kawasaki Voyager ZN1300" last week and found an '85, plus another one for a parts bike, both together for just $2400. The best one looked awfully good from the pics. He said the parts bike only needed some reassembly to run good again. A lot of these bikes run about $1500 to $2400 if they are half decent.
These bikes are out there, and some are really great. You just need to find them and be willing to travel a bit to pick it up.
To answer your main question, though, the cooling system seems to be the weak link with these engines, and after 30 years are a real concern. Riding in slow traffic, or just around town for a while will usually show how bad one will heat up. Look in the top of the radiator fill to see what kind of buildup there is.
On both of my recent buys they had clutch slippage issues due to someone adding the wrong oil. At 30 years old most will have rust in the tank, so a good cleaning will be needed, as well as a new filter. A flashlight will tell you what condition the tank is in. Also, look around where the fairing mounts connect to the fairing. A lot of these bikes have serious cracking. it can be fixed with some good fiberglass work, but that's a LOT of work! Check the "accordion" boot where the right-hand swingarm leg meets the back of the engine for cracks on the edges of the folds. If there are cracks, there will be a lot of rust inside the swingarm and on the driveshaft, and possibly a lot a wear on the splines on the driveshaft. Just consider that new wheel bearings will be needed after 30 years, too! Check the steering head bearings for tightness. When sitting on the bike with the engine not running, grab the front brake and rock the bike forward and backward to listen for any clunking in the triple-tree bearings. If you do hear some, be sure it's the bearings and not the fork bushings that can also be badly worn.
There are many other issues that CAN happen, but these are the common ones I have run into! And, for some reason, very few Voyagers have a single key for all of the locks! My '85 had four different keys to unlock everything! Most better lock shops can correct this!
Believe it or not, I bought both my '85 and my '84 from dealers on eBay! Most of those salesmen are just too young, and are only familiar with current machines, so they aren't knowledgeable enough to realize what these bike are! Try to get contact info for the PO if you do buy from a dealer. I lucked out and got to talk to the original, and only, owner of my '84 and found out why it looks and runs like new.
Good luck and take your time.