I'm not an expert, but I'd like to venture a response.
In classical logic, "this statement is always false" is equivalent to "this statement is false" because there's no intermediate truth value -- "true" in classical logic means the same thing as "always true".
There's more nuance here under the model theoretic interpretation of such statements:
A sentence is consistent if it is true under at least one interpretation; otherwise it is inconsistent
A sentence φ is said to be logically valid if it is satisfied by every interpretation
The particular statement you selected, "this statement is always false" is undecidable. That is, while the logic asserts it must be either true or false, there doesn't exist a procedure to resolve it, as Goedel demonstrated with his Incompleteness Theorems. In Tarski's Hierarchy of Truth, it would flip-flop between truth values. Note that these undecidable propositions have a lot more baggage to unpack in the difference of grammar (all statements are either true or false) vs. proof (our ability to figure out whether a given statement is true or false from axioms).
If you didn't intend for an undecidable proposition, then the negation of a tautology could be a good example of "always false" in classical logic.
In modal logic, we have more truth values and thus more room to maneuver. In Kripke Semantics, for example:
"possibly A" is defined as equivalent to "not necessarily not A"
If you intended to ask, is "this statement is false" undecidable in both classical and modal logic, then in general I think the answer is yes.