Yes, in the past there has been issues with drugs and prostitution but now it is something that is under control and less likely to rise again. St Kilda is fine for a family as there are many families who live in this area. As I said, St Kilda has always been a great culture and people are very friendly so there is no need to worry. Hope I helped! :)
St Kilda is known for it's great vibe and culture. At night time it is very busy so there is no dark, quiet streets that should worry you. There is sufficient access to public transport as well so it is easy to get to and from your home. If you want an area that has a great nightlife but also beach culture then St Kilda is great for you.