This is covered by Proposition I.1.16 on page 22 of Lindenstrauss-Tzafriri (in the old Springer Lecture Notes 338 edition):
Let the matrix $A = (\alpha_{i,j})$ represent a bounded linear operator $T$ from $X$ to $Y$ with unconditional bases $\{x_i\}$ and $\{y_j\}$. Then the diagonal operator of $A$ (i.e., the matrix $(\delta_{j}^{i} \alpha_{i,j})$) also represents a bounded linear operator $D$. If the unconditional constants of $\{x_i\}$ and $\{y_j\}$ are $1$ then $\Vert D \Vert \leq \Vert T\Vert$.
This result certainly is in the later editions of that book—it enters in various proofs later on—but Google doesn't let me check at the moment.
Added:
For the benefit of the readers, let me quickly summarize the relevant definitions—I'm using the slightly awkward choice of letters by Lindenstrauss-Tzafriri:
A sequence $\{x_n\}$ in a Banach space $X$ is called a Schauder basis if every $x \in X$ can be written uniquely as $x = \sum_{n=1}^{\infty} a_n x_n$.
If $\{x_n\}$ is a Schauder basis then there is a projection $P_n: X \to \operatorname{span}{\{x_1,\ldots,x_n\}}$. This projection is continuous and the basis constant of $\{x_n\}$ is $\sup_n \|P_n\| \geq 1$.
A series $x = \sum_{n=1}^{\infty} x_n$ in a Banach space is said to be unconditionally convergent if for every permutation $\pi$ of the positive integers the series $\sum_{n=1}^{\infty} x_{\pi(n)}$ converges.
A (Schauder) basis is called unconditional if the representation $x = \sum_{n=1}^{\infty} a_n x_n$ converges unconditionally. One can show that with an unconditional basis one can omit summands and arbitrarily change their signs and rearrange their order without affecting convergence (hence the term). See e.g. M.M. Day, Normed linear spaces Chapter IV, §3 and §4.
Let $\{x_n\}$ be an unconditional basis. If $I$ is a finite set of natural numbes then write $P_I$ for the projection onto the span of $\{x_i\}_{i \in I}$. The unconditional constant of an unconditional basis is $C = \sup{\{ \|P_{I}\|\,:\,\emptyset \neq I \subset \mathbb{N}\text{ finite}\}}$. If $C = 1$ then the basis is called $1$-unconditional.
For your convenience I'm reproducing a picture of the easy but very slick proof:

Added: Lindenstrauss and Tzafriri attribute the result to A. Tong, Diagonal submatrices of matrix maps, Pacific J. Math. Volume 32, Number 2 (1970), 551–559 where you'll find more general and more specialized variants. For a "detailed study of some related and deeper questions" they refer the reader to S. Kwapień, A. Pełczyński, The main triangle projection in matrix spaces and its applications, Studia Mathematica, Vol. 34 (1970), 43–68. See here for the full text of that volume.