In general passports were mandatory in the 1960s. For UK passports, records of issue were mostly destroyed after 11 years.
In addition to a passport, the emigrant would often have to have obtained a visa (or similar document) from Embassy of the country they were migrating to, authorising them to settle there. Finally the immigration authorities in the new country may have a record of the arrival (though whether such records for the 1960s would be open to the public I can't say).