It does depend on the structure of the problem. Lets see what we are trying to achieve here.
In your example, we want to minimize a function $f(x)$ subject to the constraints $h(x) = 0$ and $g(x) \le 0$.
When we introduce the multipliers, we form the expression
$$
L(x, s, d) = f(x) + s\cdot h(x) + d \cdot g(x).
$$
I think of the optimization as a game with an opponent: I control the original variable $x$ and I want to minimize this expression (In fact I want to minimize function $f(x)$ subject to the constraints, but the game guarantees that this is equivalent).
The opponent controls the introduced multipliers $s$ and $d$ and is trying to maximize the expression.
The way we design this game is so that every time I violate the constraints, the opponent can win by making his objective infinite! In particular, if I select $x$ for which $g(x) >0$, then the opponent can select $d \rightarrow \infty$ and drive $L(x,s,d)$ to $\infty$.
On the contrary, when I choose $x$ that does not violate the constraint, the opponent should not be able to drive $L(x,s,d)$ to $\infty$. That's why we restrict the opponent to only use $d \ge 0$.
Note that if we violate $h(x)=0$, then the opponent should be able to win irrespectively of whether $h(x) >0$ or $h(x)<0$ and that's why we put no restriction on multiplier $s$.
If the original constraint was $g(x) \ge 0$, then we could form a different expression: $L(x, s, d) = f(x) + s\cdot h(x) - d \cdot g(x)$, or we could keep it as it was requiring that $d \le 0$.
Finally note that if the original problem was to maximize $f(x)$, then you can always think of it as minimizing $-f(x)$.
