I will assume that the coefficients $a_{kj}(t), t\geq 0$ are all real.

The expression under the integral is what is called the initial growth rate of the matrix $A(t)$ with respect to the $\infty$-norm. Another standard term for this is the matrix measure of $A(t)$ with respect to the $\infty$-norm. Using standard facts about the initial growth rate the desired result follows.

The initial growth rate of a matrix $A \in \mathbb{R}^{m\times m}$ with respect to a norm $\|\cdot\|$ on $\mathbb{R}^m$ and the associated induced norm on the space of matrices may be defined as follows: It is the infimum of the numbers $C$ such that for all $t\geq 0$ we have
$$ \|e^{At}\| \leq e^{Ct} .$$

It is well known that the initial growth rate $\mu(A)$ of $A \in \mathbb{R}^{m\times m}$ with respect to $\|\cdot \|_\infty$ is precisely
$$ \mu(A) = \max_{k=1,\ldots,n} a_{kk} + \sum_{j\neq k} |a_{kj}| .$$

It follows that for a linear time-varying differential equation
$$ \dot{x}(t) = A(t)x(t)$$
with transition matrix $\Phi(t,s)$ we have for all $t \geq t_0$
$$ \| \Phi(t,t_0)\| \leq \exp \left( \int_{t_0}^t \mu(A(s)) ds \right) .$$

So with your assumption we have that for all initial times $t_0$ we have that $\| \Phi(t,t_0)\|$ is bounded in $t$. This is Lapunov stability of the zero position for all initial times. Note that the assumption does not imply uniform stability-

References for facts about the initial growth rate or matrix measure are

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Hinrichsen & Pritchard, Mathematical Systems Theory I, Springer, 2005
Vidyasagar, Nonlinear Systems Analysis, SIAM, 2002
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