Connecting the two systems outright probably doesn't make sense. It seems to me like it would make controlling them difficult, you'd have return air problems unless (and maybe even if) you also connected the return supplies, you'd need relatively big vents to make a difference, and because the systems are sized to handle half the house you're potentially overworking them by having them each now handle the entire house.
One big system
Replacing both with one larger system may make more sense -- especially when eventually you have to replace one or both anyway.
Independent recirculating vent
The intermediate alternative that comes to mind for me is to build something to recirculate air, like a single large vent that connects the two floors (but is not connected to other HVAC vents) with a fan (though it's not immediately obvious to me if the fan should blow up, down or be reversible based on cooling season). The open space from your stairs would essentially be the 'return' for this system, as the goal is to simply try to circulate air between the two floors to more evenly balance the temperature.
At increasing levels of complexity, this could be controlled manually by a simple switch, on a timer, or based on temperature differential (probe upstairs and down, and fan runs if difference passes some threshold). You'd probably have to custom build the last one.
Whether this makes enough difference to justify the cost (both install and ongoing cost to run the fan) I could not say.