  Let $\mu$ be a finite positive Borel measure on $[0,1)$ and
$f(z)=\sum_{n=0}^{\infty}a_{n}z^{n} \in H(\mathbb{D})$. For $0<\alpha<\infty$,
the generalized Ces\`aro-like operator $\mathcal{C}_{\mu,\alpha}$ is defined by
$$ \mathcal
{C}_{\mu,\alpha}(f)(z)=\sum^\infty_{n=0}\left(\mu_n\sum^n_{k=0}\frac{\Gamma(n-k+\alpha)}{\Gamma(\alpha)(n-k)!}a_k\right)z^n,
\ z\in \mathbb{D}, $$ where, for $n\geq 0$, $\mu_n$ denotes the $n$-th moment
of the measure $\mu$, that is, $\mu_n=\int_{0}^{1} t^{n}d\mu(t)$.
  For $s>1$, let $X$ be a Banach subspace of $ H(\mathbb{D})$ with
$\Lambda^{s}_{\frac{1}{s}}\subset X\subset\mathcal {B}$. In this paper, for
$1\leq p <\infty$, we characterize the measure $\mu$ for which
$\mathcal{C}_{\mu,\alpha}$ is bounded(or compact) from $X$ into analytic Besov
space $B_{p}$.
