You have had to wait a month for an answer. I guess the real expert Sashka78 has had more important things to do. Anyway I will give you an answer. Your surname Basiuk in Ukrainian Cyrillic letters is Басюк. When I googled it I got ten pages. I would guess it is not the most common Ukrainian surname but anyway there are both famous and ordinary people who are named Басюк. I found for example a dancer, a coach, a military historian and a 12-year-old girl who had ICQ.
What it means? It means bass player. The root Бас means bass. The suffix -uk or -iuk is a diminutive generally used to form patronymics, and it appears in eastern Poland, Belarus and Ukraine. So usually you can take off -uk or -iuk and render the name "son of _," and usually the first part of the surname is clearly a first name or occupation. Here is an example of the word басюк used as bass player and not as a surname. Source:
Просмотр темы - Барвы: Claus tro phobic Party (05.09.2007) • Metal.Боярка.name: Форум
Quote: "Грали в 4 людини - вокаліст, гітарист, басюк, друммер." (4 people were playing – a vocalist, a guitar player, a bass player, and a drummer.)