Polchinski "String theory" (B.2.8) The $\mathcal N = 1$ algebra
\begin{equation}
\{Q_\alpha,\overline Q_\beta\} = -2P_\mu\Gamma^\mu_{\alpha\beta}
\end{equation}
is in a frame where $k_1 = k_0$
\begin{equation}
\{Q_\alpha, Q_\beta^\dagger\} 
= 2k^0(1 + 2S_0)_{\alpha\beta}
\end{equation}
or in $\bf s$ basis,
\begin{equation}
\{Q_{s_0',s_1'},Q^\dagger_{s_0,s_1}\}
= 4k^0\delta_{s_0,1/2}\delta_{\bf s'\bf s}.
\end{equation}
The matrix elements of $Q_{-1/2,s_1}$ vanish.
The remaining operator gives the raising and lowering operator:
\begin{equation}
b = (4k^0)^{-1/2}Q_{1/2,-1/2},\;\;
b^\dagger = (4k^0)^{-1/2}Q_{1/2,1/2},\;\;
\{b,b^\dagger\} = 1,\;\;
b^2 = (b^\dagger)^2 = 0
\end{equation}
We construct the representation by starting from $|\lambda\rangle$ {\it s.t.},
\begin{equation}
S_1|\lambda\rangle = \lambda|\lambda\rangle,\;\;
b|\lambda\rangle = 0.
\end{equation}
By operating $b^\dagger$,
\begin{equation}
b^\dagger|\lambda\rangle = \Big|\lambda + \frac12\Big\rangle,\;\;
S_1\Big|\lambda + \frac12\Big\rangle 
= \Big(\lambda + \frac12\Big)
  \Big|\lambda + \frac12\Big\rangle.
\end{equation}
My question is why the state $b^\dagger|\lambda\rangle$ is
$|\lambda + 1/2\rangle$?
P.S.
I think that in order to confirm it we need to calculate the commutator $[b^\dagger,S_1]$ and it is equal to $1/2$.
 A: You need to use the super Poncaré algebra, that besides the commutator $[q,q]\sim p$ there is also the commutators $[M,p]\sim p$, $[M,q]\sim q$, and $[M,M]\sim M$. Here $q$ are the supercharges, $p$ are the momentum components and $M$ the Lorentz generators.
Turns out that using Jacobi identity
$$
[M,[q,q]]=[[M,q],q]+[q,[M,q]]
$$
and that $[q,q]\sim p$ we learn that $q$ must be a spinor, so it has charge $\pm \frac{1}{2}$ under $S_0, \dots, S_{\frac{d}{2}}$ Cartan subalgebra of the Lorentz generators
$$
[S,q]\sim \frac{1}{2} q 
$$
I did not put indices up to here since all I said is valid to any dimension and notation. For your case where we have $d=1+3$, we can work with the $SL(2,\mathbb{C})$ notation
$$
\alpha=(1\equiv ++,\,2\equiv --)\qquad \dot\alpha=(\dot 1\equiv +-,\,\dot 2\equiv -+)
$$
where the first sign is the charge under $S_0$ and the second is the charge under $S_1$. The momentum is given by $p_{\alpha\dot\alpha}$ and the supercharges are $q_{\alpha}$ and $\bar q_{\dot\alpha}$ such that
$$
[q_{\alpha},\bar q_{\dot\alpha}]=2 p_{\alpha\dot\alpha}
$$
and we have
$$
[S_0,q_{\pm \,\star}] = \pm\frac{1}{2} q_{\pm \,\star},\qquad [S_1,q_{\star\, \pm}] = \pm\frac{1}{2} q_{\star \,\pm}
$$
that follows from the Jacobi identity of
$$
[S_i,2p_{\alpha\dot\alpha}]=[S_{i}, [q_{\alpha},\bar q_{\dot\alpha}]]=[[S_i,q_{\alpha}],\bar q_{\dot\alpha}]+[q_{\alpha},[S_i,\bar q_{\dot\alpha}]] 
$$
