Probably two breeding pairs as a start and then crossing their offspring to avoid inbreeding and the genetic deficiencies it will produce over time.
To answer the question about which is healthier: In theory none. Both DNA wise are the same in regards to structure. The information on how long a fin is or what colours are they plays absolutely no role in its health.
However there are other factors such as skeletal structure that an inexperienced person might not take into account. Yet though, even that might be an accident of birth so to speak and might not be an inherited genetic condition or might relate to conditions external of the fish's upbringing (ie some fish that are bought bent and when nursed turn straight again: Check the broken heart thread to see what i write of).
To sum it up: Both show and 'random bred' fish are genetically predisposed the same in regards to health / hardiness. The only difference is their upbringing and looks. Most of the time a show betta will be healthier due to it being a prized fish and considered valuable and kept in pristine conditions and a healthier diet whereas mass bred bettas will be kept to a minimum of requirements until sold.
There are two questions that arise though which have no definite answer at least for the time being:
1)From a marine biologist friend of mine i know that fish can be 'safely' inbred for around 4-6 generations. If that limit is accurate and is been exceeded what happens to subsequent inbred generations in regards to their DNA, health and structure?
2)Given the fact that nature has a way of adapting and evolving species according to their environmental conditions what would the differences of the species been, if we had two parallel lines of betta living in different conditions? One a show line kept into optimum conditions and one of random bred kept at minimum? My guess would be that the show ones will become more beautiful yet more docile and eventually will have weaker immune systems due to not facing extremes in their environment yet with a longer lifespan (as long as they are kept comfy). On the other hand the random bred will be a haphazard and quite possibly faint colouring due to lack of quality food and conditions, leading them to become hardier and more accepting on a wide variety of conditions, been harder to succumbing to decease and quite possibly breed a survival instinct that can be translated as aggression.
Unfortunately all the above is an 'educated (of shorts) speculation' of possibilities since I do not posses the facilities to go into such a research.
However it would make an excellent idea for a PHD ;)