We examine the function $F(K,L)$ that is homogeneous of degree $\lambda < 1$. Then we have that its partial derivatives are homogeneous of degree $\lambda -1 $.
For a homogeneous function $F(K,L)$ of degree $\lambda$ it holds that
$$K\cdot F_K + L\cdot F_L = \lambda \cdot F(K,L) \tag{1}$$
Analogously for the partial derivatives we have
$$F_L:\;K\cdot F_{LK} + L\cdot F_{LL} = (\lambda -1)F_L <0 \implies K\cdot F_{LK} < -L\cdot F_{LL} \tag{2} $$
and
$$F_K: L\cdot F_{KL} + K\cdot F_{KK} = (\lambda -1)F_K <0 \implies L\cdot F_{KL} < -K\cdot F_{KK} \tag{3} $$
If the cross-partial is non-negative then necessarily the second partials are negative, and also both sides in the inequalities above are positive. Then multiplying per side we have
$$K\cdot F_{LK} \cdot L\cdot F_{KL} < L\cdot (-F_{LL}) \cdot K\cdot (-F_{KK})$$
$$\implies F_{KK}\cdot F_{LL} > F^2_{KL}$$
So a sufficient condition for (joint) concavity of a production function in two variables that exhibits decreasing returns to scale, is that the cross-partial derivative is non-negative.
Note that this sufficient condition related to the cross-partial guarantees also the "first step" for joint concavity, namely, that the second-order partials are negative (they alone provide for partial concavity of the production function).