Shielding and Ground loops
Yes, it sounds like (a little confusing) you have a ground loop problem, and yes they can matter, especially when trying to measure small analog signals.  If all grounds tie back to the same outlet strip via relatively short line cords, then it would probably be OK.  However, you say that this cryostat thing (whatever that is) is connected separately to building ground, so that is obviously not the case and it's confusing therefore why you brought it up.

In general, it's good to convert analog signals to digital as close as possible to the source, then ship around digital signals.  Those are much easier to isolate, like via opto-couplers, pulse transformers, radio, etc.  In other words, a old fashioned A/D card in the computer is not the best overall architecture from a system level point of view.

However, look at the A/D card carefully.  Most likely it can be configured for single ended and differential operation.  This is a case where you want differential inputs.  The cryostat thingy may produce a ground referenced signal, but take its ground and output signal as being differential.  This will essentially subtract the ground offset from the signal before converting it.

This trick will only work up to some frequency, probably a few kHz or low 10s of kHz.  It should work pretty well in subtracting off any ground signal due to 60 Hz or 50 Hz power line return currents accross ground paths in the loop.  Sharp common mode spikes can still confuse the diff amp in the A/D and show up as noise in the final output.  It's worth a try though.  If it's not good enough, go back and convert to digital at the sensor, then opto-isolate the digital telemetry signal.

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