That being said, there are crowded neighborhoods in Tokyo that are a challenge to navigate with young ones. The city also has several large (somewhat dangerous) red light districts. These are the areas to avoid.
1. KabukichoKabukicho in Shinjuku is a large, dangerous red light district that should be avoided.
There's no reason to avoid Shinjuku itself. West Shinjuku is a relatively safe business district that has some of Tokyo's best hotels. Keep in mind that Shinjuku is the busiest train station in the world. It's difficult to navigate with young children.
2. ShibuyaShibuya has a small red light district and a clubbing area known as love hotel hill.
Shibuya is crowded (a challenge with with small children). On the positive side, Shibuya has some of Tokyo's best shopping. It's also home to NHK Studio Park — a public television studio attraction that's of interest to young families.
3. RoppongiRoppongi is Tokyo's best and worst neighborhood. It's an upscale district that's home to some of the city's most notorious nightlife. Bars in the area go all night and into the next afternoon.
The main family safe attraction in Roppongi is Roppongi Hills — a city within the city that features upscale shopping, restaurants, a large movie theatre and a well known modern art museum.
4. GotandaThere's nothing of interest to families in Gotanda (with the possible exception of budget hotels). It's a bit seedy.
5. IkebukuroIkebukuro is Tokyo's second most crowded neighborhood. It also has a largish red light district.
6. Shin OkuboShin Okubo is Tokyo's Koreatown. Many of the Korean restaurants in the area are excellent but have a bar-like atmosphere. The area is safe enough in day. However, it's very close to Kabukicho. Things can get a bit sketchy late at night.
7. UenoUeno is an old, central neighborhood of Tokyo. Its large park is jam-packed with museums and cultural institutions. It's also home to Tokyo's best zoo (Ueno Zoo).
There's no reason to avoid Ueno's many family attractions. However, keep in mind that it's not Tokyo's safest neighborhood. Ueno is known for its large homeless population. There are also plenty of gangster types in Ueno. It's generally safe in the day. However, avoid Ueno after 9 pm.