There are different kinds of “dirt fields.”
There are dead pastures that have been overgrazed, and can either have good drainage or turn into mudpits. Some of these can sprout a bit in summer, either grass or inedible weeds.
There are natural terrains in desert or forest regions that just don’t have much grass.
There are “groomed” turnout facilities with gravel or sand or hogfuel bark mulch footing, which can have good drainage or turn into wading pools.
Obviously in all these cases two things apply, which also apply to horses only kept in stalls.
The horse will need to be fed everything it needs, hay and supplements, etc.
The field will need to picked out every day unless it is really huge. A grass field can “absorb” manure over the summer. The horses won’t eat the very new green clumps of grass growing out of fresh manure, but over the winter (when the pasture is resting) the manure will break down and fertilize the next year’s grass.
If there is no grass in the field, the manure will just sit there and become effectively the new footing unless it is picked out. Mud or dry, this can be really really disgusting and is also a high risk factor for breeding worms.
I would look at the overall horse care in choosing a facility.
I would rather be in a well-run boarding barn with good feed and facilities, and groomed turnouts than in a place with marginal pasture and skinny horses.
If however you are looking at a mud pit of a dead field that was trashed by overgrazing in the summer, and if you think that this failure in pasture management is reflective of the level of care and knowledge of the whole facility, that is another matter.
Obviously good healthy well managed pasture is the ideal, but in most of our suburban boarding situations, there simply isn’t room to maintain this.