If you were to walk , it would take about 15 mins, though having a toddler may not be the best idea as its all uphill to the station and also the hotel which is right next to the station.
You could pick up the number 6 bus from the old steine which would take you to the train station or get a cab, which won't cost you much.
The Palace pier is almost opposite Pool Valley, but the walk back down to the sea is easy as its all down hill.
I would say a lovely walk between the Palace and also West pier(the one which burnt down), lots of things for kids to do, carosel, playing on the beach, shops, cafes etc through to the kiddie pool and play park just west of the west pier.
Though on a rainy day the beach obviously is not the best place to be.
To get to the beach from your hotel you'll pass through the main shopping area and the lanes are a lovely place to walk through for adults seeing all the history.
There are some lovely restaurants in the lanes and also north lanes which are near the hotel your in.