I am translating a book published in 1961. The author has been dead for more than 50 years. I would like to use the same illustrations, but do not know who to ask. The painter is dead, and he did not have a family. The publisher has been closed for a long time.
Never mind the illustrations. Making a translation of a work protected by copyright is an infringement unless one has permission. This is true in pretty much every country. Specifically it is true in every country that has signed the Berne Copyright Convention.
Many (but not all) works published in 1961 are still under copyright. The author may be dead and have no family, but someone will have inherited the copyright — the government, if no one else. The owner could sue for infringement if a translation is created, and particularly if it is published, without permission. Permission to use the illustrations might well go with permission to make the translation at all, but in any case they also would require permission.
Some of the resources mentioned in the answer by Leon Conrad may help in finding the owner and obtaining permission, or in checking if a work is still protected by copyright.
You might get additional useful answers to this on the Law.SE site, where copyright issues come up often.
Copyright law is complex and differs across different countries.
A starting point, if you haven't used it, is the Copyright Watch centre:
The Copyright Alliance have some helpful links:
I've seen disclaimers in forewords in books which contain wording along the lines of
'every attempt has been made to contact copyright holders of works included. Ommissions unintentional. Please contact the author or publisher if you feel a copyright breach has occurred and we will do our best to rectify it.'
If you are a member of an organisation such as the Society of Authors, UK (https://www2.societyofauthors.org/) you may be able to seek advice through them or have a friendly copyright solicitor recommended who is used to dealing with these kinds of queries.