Generally, you should enforce as many relevant rules as possible in the database, otherwise they will be violated. Enforcing integrity with declarative constraints in the database is far more simple, reliable and efficient than doing it procedurally in some application or UI layer. It can even make queries faster, depending on how adept your DMBS is at semantic optimization.
More specifically, you seem to be asking about constraints on surrogate keys, not necessarily primary keys. Whether a key is primary or not has little relevance for anything at all, so I suggest you edit your question.
Speaking generally again, constraints on keys (or key columns) are of course valid and useful. Natural keys often have a specific format which should be enforced in the database, preferably as a type constraint, if your DBMS supports user-defined types.
But a surrogate key usually has no meaning or structure in and of itself—its only requirement is that its values be unique. There isn't even any logical reason for it to be an integer; it could be anything at all, though integers are often chosen for practical reasons. Thus, for a "pure" surrogate key, there are by definition no rules to enforce, apart from uniqueness and the trivial type constraint, which your DBMS already handles.
Your reasons for wanting to enforce positive-valued ids seem to be aesthetic, not based on real-world requirements. Would it actually be an error (an inconsistency) if you encountered a zero- or negative-valued id? If so, by all means add the constraint (but document the need for it). If not, don't.
One last thing: If a surrogate key value is ever entered manually (e.g. in some UI, or in ad-hoc queries), auto-incrementing integer surrogates are dangerous. It only takes a trivial typing mistake—omission, addition, transposition—to identify the wrong tuple. In such cases, your surrogate id should include one or more check digits/characters to catch most (though of course not all) such mistakes; and the validity of these ids should definitely be checked by a constraint.