Personally, I don't see how it's financially viable in most situations, but I'm guessing it's financially viable in a few situations or when shorthanded.
Although horses are expensive to "maintain", almost every ranch already has them. Feed and medical are the two largest expenses, and you're going to be providing that whether or not you are using them for mustering/herding. Horses fart and breathe, but besides that, they don't produce exhaust. I'm guessing they produce methane, but I've never measured it myself. They also produce manure that enriches the soil. Their hooves help break up the soil, which allows grasses and plants to prosper. Animals like horses, bison, and cattle are good the health of the land.
ATVs are inexpensive to fuel and somewhat inexpensive to maintain; they are not too expensive to purchase. ATVs produce toxic exhaust. Unlike horses, they compact the soil, which inhibits grasses and plants from growing. Over time, ATVs can be bad for the land, but using them on the same tracks can significantly help reduce this degradation.
Helicopters are expensive to purchase (or rent) and time-consuming to properly maintain. Fuel and maintenance costs will depend on the model. Helicopters produce toxic exhaust. Helicopters can cause loose soil to get moved, but this is usually not a problem, as the net effect is usually minimal.
There are two situation where I can see a helicopter being an advantage over horses and ATVs:
When you are shorthanded. As the size the of the herd increases, and the size of the range increases, you need more cowgirls and cowbows on horses or ATVs to manage the task. A helicopter could help in shorthanded situations.
When the range exceeds that which horses or ATVs can travel while having the feed/water or fuel they need. For horses, this depends entirely on the breed, condition, and age of the horses and the breed, condition, and age of the herd. If appropriate and sufficient food/water will not be available for the horses, but will be available for the herd, horses may not be viable. I have never seen this happen, but I could create a contrived example to make it happen, and it's possible such real-world examples exist. For ATVs, I think it's more obvious.
Meaning absolutely no disrespect to John K whatsoever, I will politely disagree with two of the reasons he posted:
Speed. The limiting factor of moving a herd is typically the speed of the herd itself. A helicopter does not move a herd faster (unless you are doing something inappropriate with the helicopter and causing the animals undue stress).
Stragglers. Skilled ranchers don't leave stragglers. Yes, mistakes can happen (such as an animal stuck in the brush and not being seen), but they are very rare, and a helicopter could make the same mistake. Since a helicopter pilot cannot hear an animal crying, a calf calling its cow, or its cow responding, I think the risk of stragglers would actually be greater with a helicopter. It's possible a helicopter provides some sort of advantage here that I'm not thinking of, but I have never flown a helicopter for this purpose, so I am not personally aware of any. A helicopter could sweep a large area after moving a herd, but I've always used fixed-wing aircraft for periodic sweeps of large areas, not a helicopter.
John did mention cost, and I wonder if this is a primary factor. Cost will depend on many factors and variables, so we don't have a clear picture. I can create theoretical situations where employing helicopters would be cost-efficient, and those in which they would not be. But since there are at least a dozen different variables at play, it would really depend on the exact situation.
