We visited the Tenement Museum a couple of weeks ago. Make sure to make reservations and buy tickets before you go to NYC. We bought tickets about 6 weeks earlier for the one-hour "Hard Times" tour. I heard other tourists in the office trying to buy tickets the day we were there and everything was full.
Suzanne was our guide and she did a phenomenal job of explaining the lifestyle, the economic environment,and the city life of NYC in the lower East side in 1865 and in 1910. We toured two actual apartments that were occupied during two different time periods. She explained what life would have been like for these families and showed us pictures of them and their descendants. My daughters especially liked it because she was telling us about actual people that lived there (not speculation of what it would have been like).
Tickets for adults are $22 each and well worth it. This was one of the most interesting and meaningful sites we saw in NYC. The walk through Chinatown from the Spring St. subway station was interesting too! (We should have gotten off at Delancy St. to be closer.)