Ecologists define a "wildfire" as one which occurs naturally. Generally these fires are stared by lightning. Wildfires are essential to some ecosystems; stopping these would result in ecosystem damage and extinction of certain fire-adapted species. The science of fire biology has developed to study this very question. There are many places where wildfire is a naturally occurring abiotic part of the ecosystem. In such places species have adapted to not only survive fire, but sometimes to actually need fire to complete their life cycles. Fire is essential to the maintenance of prairies, of pitch pine, longleaf pine, and sequoia forests, and also of chaparral.
The issue for humans arises when people choose to build homes and other buildings in fire-adapted ecosystems, and then attempt to protect the structures. Suppressing fire in a fire-adapted ecosystem not only creates negative impacts to the native species, it also allows large buildups of dry fuel to accumulate, so that eventually a truly large and very damaging fire becomes inevitable.
We cannot protect our forests by building our homes in them and then expecting the forest's ecology to change to accommodate our presence.
This is a great question. There are a few important points to be made.
First, there are some wildfire that need to be stopped at all costs. This is so when the life of people and their way of life is threatened. Recently, this happened in New York State and even New Jersey (last week and this week). The firefighters valiantly stopped the spread with minimal loss.
Second, when fires are started by carelessness or worse intentionally, these too should be stopped as well. In these cases there is a risk that people will get hurt, because they do not happen naturally.
Third, there are some wildfires that happen naturally. When this happens, it is difficult to know what to do exactly. The reason for this is because wildfires are a way nature gets rid of unnecessary excess and creates for life to take place in time. Ironically, these wildfires promote life and even wildlife. These ecosystems are called fire dependent ecosystems. In these cases, we should leave the decision to the experts.
Finally, I will add a link to Smokey the Bear.
In his Gaia hypothesis, Lovelock (1974) cited wildfires as a mechanism maintaining O2-CO2 balance between biosphere and atmosphere. It is pertinent to note here that geological evidence of wildfires, preserved in rock successions, far outdates the advent of anthropoids. Charcoal beds that are held as proxies for wildfires in rock record are recurrent, since Silurian. These event beds are temporally correlated with high biomass and corresponding high oxygen content in the atmosphere.