We don’t know.
Easiest question ever!
Scripture has provided us with very little evidence of St. Joseph’s life much less his death. He has no spoken words in the Bible and God only speaks to him in dreams. We assume that Joseph is dead because we don’t see him at the foot of the cross with Mary at Jesus’ crucifixion.
It was also assumed that Joseph was much older than Mary and so in an age where people didn’t live that long Joseph would have died long before Mary. Life expectancy was about 40 years old at the time and so if Jesus was 33 when he died then Mary was well into her 40s, a very old woman for her time! Joseph would have to have been in his 50s and it is unlikely that he would have lived that long.
Still we take comfort in the fact that Mary and Jesus were taken good care of by Joseph. Why? Because they lived as long as they did! Surely Joseph took good care of this family if they were able to escape all the tragedies that could befall them in this age.
So Joseph has a hidden life, a life that goes unseen by the gospel writers and that should be an inspiration to all of us who do work behind the scenes and often feel unnoticed.