First, I don't remember seeing a question like that on the ASTB. Second, that's a horrible diagram.
To me, it looks like it's a simple model of a tower crane, meaning that the 'anchor' is not attached to anything - it's there to balance any loads on the right horizontal support. The term 'anchor' does not necessarily mean a rigid attachment. The answer depends on whether the anchor is a just a weight, or a rigid point of attachment for the horizontal support. It also depends on whether that string can slide across the top of the vertical support, or if it's attached there.
Here's a quick list of answers for an attached/non-attached anchor, and a sliding/non-sliding string:
Attached/Sliding: B. B causes more tension because it bends the horizontal support and stretches the string, A can't pull the string because the anchor supports it's weight, and the left horizontal support can't move.
Attached/Non-sliding: C. The tension at both weight locations is 0 because the anchor supports A, and the bending caused by B doesn't pull the string on the left side.
Non-attached/Sliding: Unanswerable. It depends on the relative weight of the block and the anchor, and the relative distance of A and B. Both positions will cause more overall bending of the support and stretching of the string, but you can't tell how much without numbers.
Non-attached/Non-Sliding: A. B won't affect the tension, because the string isn't free to slide. A will increase the weight on the left horizontal support, causing it to bend more.
Since the word "string" is written only once, I guess you're safe assuming that it's one continuous string, and is free to slide. You can also assume the question is answerable, so the anchor probably is a rigid attachment. So, the answer is probably B like Sinatra said. I guess I made the whole thing more complicated, but I really hate bad diagrams and questions that give incomplete information.