Been to many TdF stages and a couple of Giro ones.
With kids, it depends on how into it they are!
Time trials are great as you get several hours of action. Team TT's are even better to watch!
A stage finish/mid point can be an anti climax.For a good spot you wait several hours for only seconds of action. I'm ok with that, my family a bit less so!
A nice mid spot can be good if you get the weather and make it a picnic. Especially if its not a busy spot. You can get loads of freebies from the caravanne, which my kids loved - even the "rubbish" stuff, they still liked running after it and finding it.
Stage starts can be great. Especially in smaller towns, you can get close enough to the riders for photos and autographs and a close look at the buses and bikes.
I like to ride a bit of the stage beforehand too. Easilly done out in the country but even in towns and cities.
I once warmed up with Team Columbia in the car park at Harland & Wolf in Belfast. I then followed a Team Sky recce car out and followed it to the edge of the city before going back into the centre to find a good spot to watch. Its a real thrill to ride bits of a course when there are hundreds of folk waiting for the race. They will cheer you on! In Yorkshire there were literally thousands of people cheering (and helping with a wee push) me on up Buttertubs.
A base thats near a finish and a start, maybe including a TT would be my first thing to look for. You can then try a couple of different ways to enjoy the action and see what works for all of you. If you make it a holiday, take your bike and ride at least part of a stage yourself. Don't ever ask a steward if you can hop a barrier to get on a course, just do it!
The one thing I've not done is a mountain stage. I'd love to, but I'm not sure its done easily with the family in tow unless you are in a camper van. Even then, you probably have to get up the mountain a few days before the stage.