Taxi is by far easiest and quickest.
The closest bus terminal is Halıcıoğlu and you have go take an out-of-the-way bus route from the Grand Bazaar terminal that includes a transfer.
You can take a ferry. Go to Eminönu and walk past its ferry station, the bridge, the plaza and the bus terminal to the small ferry station for the Golden Horn Ferry that terminates at Eyüp. Ferries are reasonably frequent. One you pass the floating submarine on the right (you can't miss) get off at the next stop (Sütlüce). Then either get a taxi or walk inland to the main road, turn left and follow about 4 km to Miniaturk. The submarine is on the site of the Koç Museum of Industry and Transportation which I highly recommend--it's an EXCELLENT museum geared to both children and adults. From the ferry station it's a short walk to the right on the main road.
If you're with family I definitely suggest taking the ferry. The final stop (Eyüp) is very near the cable car up to Pierre Loti (an observation point/famous tea house) by way of a wonderful square a notable mosque (Eyüp Sultan) and an authentic Ottoman wood home neighborhood that you can't miss if you take the nice walk down from Pierre Loti.