The 610C is from before triax inputs, so replacement is easy. Any RF coaxial cable having a decent (no holes) braid screen will do. RG174 would be fine, but RG59 (75 ohm) or RG58 (50 ohm) are not so good because their braid has holes. I'd replace the input connector as well. Not because it was a poor original choice but because it has probably got dirty (leaky) over the decades. That's why electrometers should have protective caps over their input connectors (and also over their associated cables). Be aware that the longer the input cable, the higher the shunt capacitance and noise gain of the meter. Keep it short.
All electrometers are horribly sensitive to capacitively coupled hum (hence that need for a good screen) and you should always have an oscilloscope on the pre-amp (chart recorder) output to check that you have minimised hum before believing any DC measurement. The best electrometer measurements are made inside earthed biscuit tins connected by a short cable to the electrometer.