Yes your definitly worrying TOO much. Ayia Napa had a bad reputation years ago but it hasn't been like that for a long time. It's very family friendly and you can still have a great night out without even touching bar street and the square if you wanted.
They got a new lady mayor several years ago and she was determined to clean the place up and restore it's reputation. She's managed this really well as the place is a popular mix of people now from families to hard core clubbers and everything in between
The police are heavily present in bar street and the music has to be turned off in the pubs by 2am which they heavily enforce.
I've been going for the last 20 years at least twice a year and have seen Napa through all its ups and downs but I've never thought twice about going and I always feel safe.
Go and try it especially some of the fabulous restaurants that are available, I promise you'll go back once you've visited