Knowing the make and model of the RV will help in getting a better answer. That said, most, if not all, motor homes do charge both sets of batteries from the alternator while underway, UNLESS something is not functioning correctly.
Welcome to the RVForum. Stu gave you good info. All motorhomes that I'm familiar with have charging systems that enable the alternator to charge both the house and chassis batteries simultaneously, but different manufacturers often design their charging systems somewhat differently.
Our system, for example, is controlled by a Battery Isolation Manager (BIM) and it doesn't allow the alternator to charge the house batteries until they get down to 80% SOC, and only then if the chassis batteries are at, or near, full charge. There are other charging systems that have different charging parameters.
If your alternator isn't charging your house batteries at all, your battery isolation solenoid may not be functioning probabaly, there could be a wiring problem somewhere or...? It depends on how your system is designed.
On most motor homes. if not all. teh answer is by RUNNING THE ENGINE. it's automatic.
Most all motor homes. Every one I've seen (Possibly not some "Conversions" but every one I have seen built as a motor home from the chassis up) have a device called an ISOLATOR. generally a solenoid or "Big Relay" it connectes the two systems together when the engine is running so the alternator charges both batteries.
HOW TO TEST (2 person job)
Hook a volt meter to the House (Cabin) battery. note voltage. have partner start engine (leave in park) and rev to "high idle" (Say 1000-1500 RPM) for one minute while you watch the meter. Did it jump up a bit? If so you have an automatic isolator that does what you ask.
The reverse: (Charging chassis battery or engine battery on shore power)
Some do that as well. these have what we call a B.I.R.D. System (Biderictional Isolator Relay Device or Delay) same test only only one body needed. Measure chassis with all off unplugged.. Plug in and watch chassis battery volt meter for a couple minutes. Should jump up to 13. 6