Welcome to The Answerbank.
Given the timing of your post (0349 GMT), I suspect that you might be in the USA. This website is based in the UK and I doubt that anyone here has even heard of those bears (whichever way you spell their names), since the books and the TV series seem to be an entirely American thing.
However, despite the 'conspiracy theory' over parallel universes (
http://www.avclub.com/article/how-you-spell-berenstain-bears-could-be-proof-para-223615 ) all of the evidence shows that the 'a' spelling is the correct one.
For example, all web references to the authors' surname give it as 'Berenstain', e.g.
http://www.nytimes.com/2005/11/30/books/stan-berenstain-cocreator-of-those-fuzzy-bears-dies-at-82.html?_r=0
Further, a search of the online catalogue (or 'catalog', if you must!) of the Library of Congress finds no entries whatsover for 'Berenstein Bears' but 383 for 'Berenstain Bears':
https://catalog.loc.gov/vwebv/search?searchArg=Berenstain+Bears&searchCode=GKEY^*&searchType=0&recCount=25&sk=en_US
(A search of our own British Library catalogue shows that the books were never published in the UK, with either spelling).
So I'm prepared to put my money on the 'a' spelling ;-)