Leave the batteries hooked together in their 12V series-parallel connection for charging.
For winter storage, there's generally little need to remove the batteries to your basement. The cold will not hurt the batteries in the least so long as they aren't discharged. If you charge them fully and disconnect them (so there are no parasitic loads), they will not self-discharge too much over a New York winter as self-dischrage rates are quite temperature dependent and much lower at cold temperatures. If you have solar or power available, so much the better to keep them charged up in the RV.
What charger or converter to use depends somewhat on how big a generator you have. For boondocking, the general goal would be to get as much charge into the batteries as quickly as possible, which boils down to having a charger or converter with a sufficiently high voltage output that maximizes the (apparent) power the generator can produce. Many converters have rather poor power factors, so the apparent power is somewhat higher than the actual power consumption.
(If you aren't familiar with apparent power vs. real power, here's a quick rundown. "Apparent power" is the product of the RMS current and RMS voltage of an AC load, the two measured independently. "Real power" is the actual rate of energy consumption, computed by integrating the instantaneous power over time. Apparent power will be higher than real power if the current waveform is not a sine wave in phase with the voltage waveform. The ratio of real power to apparent power is the power factor, which is always less than 1 and often expressed as a percentage.)