In the Harry Potter series, why do they select Muggleborns to also learn magic at the Hogwarts School of Witchcraft and Wizardry and other schools in the wizard world?
Simple: because they possess magic. If they don't learn how to control it, they will probably end up hurting people, and more importantly, they might use magic in front of a large group of people, which would expose the existance of magic (and probably start the search for other magicians). Think X-men, but with magic.
They could also just kill them, but that would probably lead to outrage in the wizarding community, since a lot of wizards don't hate Muggleborns.
They teach anyone who shows magical ability. It is the responsibility of the schools to do so to facilitate the ministry in 'controlling' magic. It means that everyone is able to control their own magic whether born of wizards or muggles and besides as Hagrid would say.
And it's codswallop to boot. "Dirty blood." Why, there isn't a wizard alive today that's not half-blood or less. More to the point, they've yet to think of a spell that our Hermione can't do. Come 'ere. Don't you think on it, Hermione. Don't you think on it for one minute.
Hagrid: The Chamber of Secrets (Movie)
In addition to the answers already given, there is also the consideration that the Wizarding world is a small one. If they didn't train Muggle Borns in their world, it's probable that they wouldn't have had enough people to sustain a civilization, let alone a population.
There wouldn't be enough people to fill the positions of administrators, teachers, technicians, and craftsman that ran the businesses, governments, schools, etc that are seen in the Harry Potter books. They would lose knowledge and capability from generation to generation due to deaths of older wizards and lack of new ones to train. There probably also wouldn't be as much innovation since the Muggle Borns often brought a different perspective and insight to magic and its practical applications.
Without a stable and growing population, it is also doubtful that they would have been able to maintain the economy and wealth that are seen in the books. Who would buy all of the little doodads and candies? Who would have the money without enough jobs to pay them a wage? Without profit and wealth being generated, how many businesses would last long enough for all of those doodads and candies to actually exist? And without the businesses, where are the jobs to pay people to be able to buy stuff?
There also wouldn't have been the strong ties between the Wizarding and Muggle worlds that are seen in the books, since a pure Wizarding wouldn't have the Muggle knowledge or consideration to interact well with the Muggle governments.
Lastly, with a lower population, Wizards wouldn't have been able to maintain a position of strength in regards to the other magical creatures. Not just in terms of political dominance as with the goblins, but also in terms of safeguarding and controlling the dragons and other beasties to preserve both worlds. Would the Inqusition and witch trials have been ended, or forgotten, if there were still monsters roaming around destroying and controlling the Muggle world?