There was an article published (MCN?...) a few years ago, which discussed the practice of using ever widening tires on the back end of bikes. The upshot was that the author had been using wider tires, thinking it would give better stability; he was afraid that narrow tires, like what the manufacturer originally spec'ed out for the bike, would fall into turns/leans, and he was afraid to try it. Long story short, he finally put the specified, narrower tire on his older bike: it never fell into turns, and it handled even better than it had with the excessively wide tires (tires wider than OEM spec's)!
Check the wheel bearings, chain tension, rear wheel alignment, front and rear suspension (fork fluid ever get changed?), etc. What kind of tire, how many miles, how deep is the tread? Are they more than three years old? Are they properly balanced? If it is a tube tire, is the tube alright, bead seal intact? Do both tires have proper air pressure?
I would suggest going over the suspension system, front and rear. My wife and I rode two-up, combined weight of around 400 lbs. plus luggage, on our '79 K model, with OEM suspension (was tired, but it never squirreled around on us), as well as Progressive Fork Springs, and 12-Series/412 shocks (can't remember which one it was), which made it ride like a modern bike (worth every penny!). We rode 1,935 miles on the OEM suspension, touring, without issue.
Are the rear Shocks leaking oil? They were sealed, maintenance-free units if OEM, but they do wear out, over time, and with miles. Check their function to see that they each react the same. Check the spring settings on both rear Shocks: are they set the same? If not, correct it -- this can make the entire bike unstable. Check the front Fork Springs, to make sure they're the same length, and that they have equal amounts of fluid in them (leaking seals? fix it ASAP). Check to make sure they have the same spacers installed, if any.
Sometimes the source of the problem will project the symptoms to another area on the bike. That is why I suggest checking front suspension, as well as rear. Check the bike over thoroughly, no matter what. Riding two-up, you are responsible for the life of your passenger. You owe it to them to be as safe as possible -- their life, literally, depends upon it.
Had a 40 MPH low-side last Fall when the rear inner tube blew out. I had my youngest son on board, age 24 now -- he had some minor road rash, I was unharmed, but it could have cost me his life. It was a factory defect in the tube, according to my mechanic, so it wasn't possible to prevent, or anticipate it.
Lot's of things to look into. I wish you the very best of luck. Cheers!
1979 CB750K (sold, 2012, but not forgotten)
1983 Kawasaki 440 LTD Belt Drive (sold, 2011)
1993 Kawasaki Voyager XII