First, know your kids. Honestly evaluate their ability to handle long days, lengthy flights, significant time zone changes and getting up early - regardless of their age. Be aware of what their interests are and what's right for them. Based on this, determine whether a Mediterranean cruise is a good choice for your family. If so, plan your cruise around your family's interests, goals, and travel profile.
Second, get your expectations in line. It's important for parents to be aware that taking a Mediterranean family cruise isn't the same kind of vacation experience as a Caribbean family cruise. Some tour-packed segments of the cruise may not be relaxing. There's a 6-hour (or more) time difference. Many of the ports are over an hour away from the major cities or attractions, making shore excursions very lengthy for younger children. Also remember that your family Mediterranean cruise experience isn't going to be a replay of that romantic couples anniversary cruise you took to Italy years ago.
Finally, pick the right time to visit. August is the prime vacation month in Europe. If you cruise during August, attractions are going to be at their most crowded and it could be scorching hot. Plan accordingly.