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Patterned assumptions seem to need to match the ConditionalExpression's condition exactly to work out for some cases.

The ∈ Reals-assumptions do work as you gave them, while the inequality Subscript[s,_]>0 does not, but observe the different behavior of Subscript[s,_]>=0:

Evaluating without any assumptions first:

f = 1/\[Sqrt](2 \[Pi] Subscript[s, 
   i]) Exp[-((x - Subscript[m, i])^2/(2 Subscript[s, i]))]
expr=Integrate[f, {x, -∞, ∞}]
(* ConditionalExpression[1, Re[Subscript[s, i]] >= 0] *) 

(Note, that the assumptions about h, k and m have no effect and are therefore superfluous.)

Now, bringing in the patterned assumptions in two versions:

greaterOnly = And @@ {Subscript[s, _] > 0, Element[Subscript[s, _], Reals]}
greaterEqual = And @@ {Subscript[s, _] >= 0, Element[Subscript[s, _], Reals]}

Refine[expr,greaterOnly]
(* ConditionalExpression[1, Subscript[s, i] >= 0] *) 
Refine[expr,greaterEqual]
(* 1 *)

So, the Element-part of assumptions is used in any case, while the inequality requires a "perfect" match.

Hope this helped!