Mathematica is a rewrite language. x_ means "match something and give it the name x for rewrite purposes". Subscript[x,1]_ is meaningless, because Subscript[x,1] isn't a symbol.
One way to deal with things like this is to deconstruct them and reconstruct what you want.
myfunction[var_Subscript] := Subscript[var[[1]], var[[2]]] + var[[2]]^2
The argument pattern here recognizes anything with the head Subscript. The function body deconstructs it and reconstructs using both parts of the Subscript expression.
myfunction[Subscript[x, 2]] (* 4 + Subscript[x, 2] *)
If you only want it to work when the subscript is 1, perhaps the handiest way is to put that in the argument pattern and do your deconstruction there:
anotherway[Subscript[x_, 1]] := Subscript[x, 1] + 1 anotherway[Subscript[y, 1]] (* 1 + Subscript[y, 1] *)
You can also do implement this kind of restriction with Condition, left as an exercise for the reader.
Subscript. It is evil. Maybe it ismyfunction[x_] := Indexed[x, 1] + 1what you want... $\endgroup$ModulewithPatternname. $\endgroup$