There's actually an interactive puppet show coming to JC that's great for toddlers. It started in Hoboken 6 years ago and is still really popular there. It's called Puppetonia.

Full disclosure: I'm one of the puppeteers.
I love what I do though, and think anyone with children ages 1-5 will love what we do too!

We're going to have a free open house and show this Wed the 18th from 12:00pm to 1:00pm at the Grassroots Community Space (54 Coles St).
And we'll be continuing every Wednesday (to start).

Here's a video to get a good sense for what the show's like.

This is a pretty old thread but I just wanted to toss in an update:

Puppetonia is actually coming back to downtown JC with a free open house this week (Wed @ noon, 54 Coles St).
I'm one of the puppeteers and have been wanting to find a home for the show in JC for a while now. I think we finally found a great plaace that works for everyone.

We'll be at Grassroots Community Space (54 Coles St).
Every Wednesday (to start off)
From 12:00pm - 1:00pm

For a better idea of what the show is like you can check out this video.
I'm sure you can also check some hoboken forums to see what they say about the show where it originated and is held twice daily.

I'm really excited about doing these show in my neighborhood!