Beatrixoord is easy to reach by public transport. There are buses from the central NS station all the way down the Rijksstraatweg. From the Dilgtweg bus stop it's only about 400 metres to Beatrixoord.
The trains between Groningen and Zwolle (both ways) stop at Haren twice every hour. From the train station you can then take a service bus, which will drop you off at the Beatrixoord main entrance.
More information about public transport can be found on the 9292-OV website, a link to this website you will find at the bottom of the page.