J.R. S. answered 11/11/22
Ph.D. University Professor with 10+ years Tutoring Experience
If Ba(OH)2 were insoluble, then adding OH- would definitely shift the reaction to the right. The reason being that as you precipitate the Ba2+(aq), according to LeChatelier's principle, the reaction would shift toward the products. HOWEVER, Ba(OH)2 is fairly soluble, so this would not be the case.
Not sure why someone told you it would shift to the right. At first inspection, I can't see why that would be the case. The added OH- can only react with one species, and that would be the aqueous Ba2+. As mentioned, that would form Ba(OH)2 would would simply dissociate back to Ba2+ + OH-.
Maybe someone else will weigh in and enlighten both of us.