Learn Polish: Mężczyzna nie mógł przekroczyć granicy bez paszportu. - The man could not cross the border without a passport.

Not always! When nie móc expresses a strict circumstantial inability to complete a specific action (he didn't have a passport, so it was physically or legally impossible), it takes a perfective infinitive like przekroczyć. We only default to the imperfective after nie móc when expressing a prohibition ("not allowed to") or an ongoing, repeated inability.

How would the meaning change if the sentence used the imperfective przekraczać instead?

Saying Mężczyzna nie mógł przekraczać granicy shifts the meaning from inability to prohibition. It would mean "The man was not allowed to cross the border." It could also imply a repeated, ongoing inability over a period of time, rather than a single blocked attempt.

Why is the border in the Genitive case (granicy) instead of the Accusative (granicę)?

This is the Genitive of Negation at work. Even though the direct object technically belongs to the infinitive przekroczyć (which normally requires the Accusative), the main verb mógł is negated (nie mógł). This negation "reaches across" the infinitive and forces the direct object into the Genitive case.

If this was a single, completed failure to cross the border, why is the past tense verb mógł imperfective?