Prove disprove an existence of mapping reduction between 2 sets
One basic idea is to observe that $B$ looks harder than $A$ since


we need to run a given Turing machine on one given input in order to find whether the pair is in $A$ but
we may need to run a given Turing machine on infinity many inputs in order to find whether it is in $B$.


That observation should motivate us to find a reduction from $A$ to $B$.



Suppose $(p,d)\in A$. Construct a machine that behaves as the following.

Upon an input string $s$, it checks whether $s$ is $d\sigma$, where $\sigma$ stand for a fixed letter in the alphabet.


If yes, it halts. 
If no, it checks whether $s$ is $d$. 


If no, it loops forever. 
If yes, it simulate $p$ on $d$.



It should be routine to verify the construction above is indeed a reduction from $A$ to $B$.