Even floating objects have capacitance. For example clouds which have huge capacitance and the charge equalization current ( cloud to cloud lightning) can be >> 10KA depending on the ionization path resistance. But heli's being a relatively "large bird" will conduct a fair amount of current to reach the same potential as the UHV line whether it is AC or DC. Once "bonded" the extremely low frequency 50/60Hz and mesh suits protect the operators. As an aside high electric fields of DC or ELF (extra low) have no detrimental effects on humans unlike RF from sustained arcing.
I read from ABB it costs $10k~20K to train an operator for live wire maintenance with 2 yrs prior experience at lower levels. So information is not freely available. They once relied on live hand connections since 1989 with a Faraday suit but now use the fiberglass rods with special conductors for connect and disconnect instead.. They don't specify details or differences between live servicing say 400 kVac lines vs UHVDC at 800 kVdc. I suspect servicing HVDC has the advantage of only having initial discharge/charge charge arc currents and DC leakage determined by dust and humidity, so I suspect servicing is limited when risk of rain is 0%. However AC lines may have more leakage due to capacitance to free-space of the Helicopter. Since the wavelengths are extremely long there is not much difference in electric fields, but as Dave said high gradients before contact. as there is in leakage current due to series capacitance of the rod to the body capacitance of the Heli.
More Aside notes...
EMI affects humans roughly proportional to f until you get to infrared wavelengths then it picks up again starting with UV and continues to get even more sensitive to gamma ray wavelengths. I don't have the latest peer reviewed journals on non-ionizing risks, but there may be plenty of scare-ware reports. Microwave Engineers are extremely cautious about cell tower design to minimize risks to humans from dispersion to near field ground radiation and apartment spacing. Continuous exposure to Tesla generators with high current arcs are not good. Whereas low current plasma balls are harmless. Power Line operators are extremely careful about the cleanliness of their grounding safety rods that may reduce the dielectric strength.
I recall a Popular Electronics story decades ago, that warned people about AC power panels near bedrooms causing cancer in the home. I remember hearing circa Y2K that it was "fraud" and the perp. "scientist" was charged for defrauding the Gov't of grant money. Similar studies have been done on Cows. But the huge EM wavelengths create little potential risk to humans ( pun intended) HV lines are restricted from property development in most areas due to extreme weather on safety of fallen lines from Ice, tornadoes or hurricanes. Its not the HV ELF fields you have to worry about, but rather continuous arcing fields which contain RF, which if sustained can cause red eyes or worse fried brains in the case of Tank operators during WWII with high power transmitters behind their brain causing them to pass out to safety below deck.
So how much current does it take to charge up on connection? I think that would depend on the altitude & relative humidity and the coupling capacitance of air dielectric to the surface area of the floating "bird" capacitance is the other factor. So it is not truly "floating" unless you are a really tiny bird with no rotating "wings" that create also tribo-electric DC charges.
The altitude effect is relative to ground. As the chopper aproaches ground < 100ft it's capacitance increases rapidly from "free space" values thus as V=Q/C the charge Q results in a rapid decrease in voltage as C increases. In any case the landing feet touch first to discharge the bird and I have never heard of arcing on touch-down but this another question, but this effect reduces the risk.