Jasinski = English - Jasiński = Polish - Ясінський = Ukrainian - Ясинский = Russian - Ясінскі = Belarusyn
Today there are appx 4,300+ residents spread throughout Ukraine, who carry this surname. This surname is likely rooted in Polish word JASIN/JASIEN, which means ASH TREE.
Ash trees grow and mature faster than oak trees and lumber made from ash trees does not shrink like other wood products do when they are dried. Wood from ash is very durable and strong. Ash tree usage goes back to the ancient Greeks and Romans, who used the wood to make spears and agricultural tools. So surname likely means person, who worked w/ash trees in some way. He could have been a furniture maker, tool maker (handles made from ash lumber) or even a medicine man (lilac-like leaves have some medicinal value).