It will stabilize wherever the base is anchored even as far north as the continental US. You can be pretty far off the equator and still get the thing to work.
Remember -- the CM of the system is still at GEO, and it's subject to the same effects that require GEO satellites to need OOP orbit control. The OOP thrusters would be to counter the effects of things like sun-moon gravitation, Earth non-sphericity, the effects of wind on the lower cable and solar radiation on the upper cable, stuff like that.
Not to mention the fact of needing to damp out tension and translational oscillations (which was a big deal even for the 20 km Shuttle tethered satellite).
It's a very complicated problem.