Then the question becomes: what counts as a "base" and how much of the surrounding land or other relevant resources are deemed to belong to the base?

Personally, I think land ownership claims around unmanned probes, rovers, and stationary facilities should be kept small, and claims around manned homesteads should not be more than about a mile radius or so, probably less. In the beginning, I think it's good to have policies that encourage settlements with different owners to be located close together, without anyone becoming a real-estate baron. The best land should go to those who have the ability to actually use it, not real-estate speculators.

See also: https://xkcd.com/2125/ as an example of how land claims shouldn't work.