we're with out toddler in Shanghai right now. most taxis have no safety systems (seatbelts or in some cases not even doors than latch right). there are some newer minivan type taxis (expo taxis) which look like they have better safety features, but I haven't had a chance to ride in one yet;there aren't that many so you may need to wait a bit for an open one at peak times.

a private car is your best option. The ubiquitous Builk GL8 minivan will run about 13-15000 RMB per month (the driver only get ~2000RMB/month so you may be able to work out something better). This gives you a driver from about 7am-7pm 5 days a week. They're happy to work weekends and overtime which you can either negotiate directly or pay more through the company. It's relatively cheap for what you get and you can leave your seat buckled into the car instead of having to lug it around.

Send me an IM if you'd like the name of the service my company uses. They have drivers in both Puxi and Pudong area and you can reqest a driver close to where you're staying to minimize gas and commute time if you don't want to have him just sitting and waiting for you.