For the British wizards the answer seems to be that there are no alternatives to Azkaban if you are imprisoned.
It all started when International Statute of Secrecy was imposed. It seemed smaller less guarded prisons were easier to escape and drew Muggle Attention, as you would expect with hose being to perform magic.
Once the International Statute of Secrecy had been imposed, the Ministry of Magic felt that the small wizarding prisons that existed up and down the country in various towns and villages posed a security risk, because attempts by incarcerated witches and wizards to break out often led to undesirable bangs, smells and light shows. A purpose-built prison, located on some remote Hebridean island, was preferred, and plans had been drawn up when Damocles Rowle became Minister for Magic.
Azkaban by J.K. Rowling
Azkaban provided an effective solution seemingly regardless of the crime committed.
In spite of opposition from many wizards, among them experts on both Dementors and buildings with Azkaban’s kind of Dark history, Rowle carried out his plan and soon a steady trickle of prisoners had been placed there. None ever emerged. If they were not mad and dangerous before being placed in Azkaban, they swiftly became so.
Once it started doing it job effectively only one Minister of Magic (until Shacklebolt) ever looked into an alternative, but it was too little too late. Also, there was a fear the Dementors would leave the island looking for more souls to feed on.
Rowle was succeeded by Perseus Parkinson, who was likewise pro-Azkaban. By the time that Eldritch Diggory took over as Minister for Magic, the prison had been operating for fifteen years. There had been no breakouts and no breaches of security. The new prison seemed to be working well. It was only when Diggory went to visit that he realised exactly what conditions inside were like. Prisoners were mostly insane and a graveyard had been established to accommodate those that died of despair.
Back in London, Diggory established a committee to explore alternatives to Azkaban, or at least to remove the Dementors as guards. Experts explained to him that the only reason the Dementors were (mostly) confined to the island was that they were being provided with a constant supply of souls on which to feed. If deprived of prisoners, they were likely to abandon the prison and head for the mainland.
This advice notwithstanding, Diggory had been so horrified by what he had seen inside Azkaban that he pressed the committee to find alternatives. Before they could reach any decision, however, Diggory caught dragon pox and died. From that time until the advent of Kingsley Shacklebolt, no Minister ever seriously considered closing Azkaban. They turned a blind eye to the inhumane conditions inside the fortress, permitted it to be magically enlarged and expanded and rarely visited, due to the awful effects of entering a building populated by thousands of Dementors. Most justified their attitude by pointing to the prison’s perfect record at keeping prisoners locked up.
For other countries there seems to be other prisons... or at least one that we know of.
We know that Gellert Grindelwald was locked up in Nurmengard prison when Voldemort caught up with him.