As far as stabilizing the relative phase delay of the fibers, there really isn't any perfect solution. Your first line of defense will simply be thermal control. Put your apparatus in an enclosure, and allow the temperature to stabilize before taking measurements. Avoid turning any equipment on or off, to maintain thermal stability.
If that doesn't work, you could go one step further and get some sort of active temperature control. I'd expect that to be available off the shelf.
If thermal control doesn't get you sufficient stability, then some sort of active feedback might be necessary. If you could measure thr fringe location and use that to drive a phase shifter in one of the fibers, that would really lock things down. If your experimental light is too dim for this, maybe you can do it at a different wavelength, and filter the extra light out with a band-pass filter.
Now, regarding the quantum measurement you are after: the exact physics of the process you are measuring is outside my knowledge, but I'm assuming from your phrasing that you ultimately want to measure the fringe spacing, our something like that, with a signal low enough to be in the single-photon regime.
In that case, coincidence measurements should show a stable spatial correlation even as the fringes drift, so if coincidences versus spacing if all you care about, you may not need the stabilization I mentioned.
In other words, if you have detectors spaced by an integer multiple of the fringe period they should correlate strongly, regardless of whether they simultaneously count a photon, or simultaneously do not. As the fringes drift, the count rates will vary, but the coincidence rate should (in a statistical sense) be fixed.