I agree, it's great staying in the city center. We did so in Rome (Homeaway) and Bruges (hotel points). And if money were no object my first choice would be the city center. However, hotels in the city center are pretty expensive for families. So we had to make a choice. We could stay outside the city center and spend 5 weeks in Europe, or we could stay at hotels in the city center and spend 10 days in Europe.
We live in the Suburbs in the US. We're use to driving 30 minutes daily to go places. So for my family a 15-20 train ride was nothing. Additionally, of the 6 countries we visited at least 2 (London & The Netherlands) allowed our kids to ride completely for free and all others allowed our kids to ride at half price on public transportation. We chose hotels that literally had a train or subway station right outside their door.
I would compare actual hotel cost, transportation costs, and transportation time. All this information is accessible online and in English. That's what I did. If you can swing staying in the city center you should do it. For us, being able to spend an extended time and see more countries was totally worth the 15-20 minute commute. And for my kids (and myself) the train rides were an awesome adventure by itself.
In Rome we stayed in a really nice and affordable Homeaway apartment in Trastevere recommended by someone here on the Rick Steves forum. If I could get comparable deals like that more often that would definitely be my first choice.