California, Los Angeles: Baby2Baby collects all things baby for low-income kids age 0-12 and has multiple dropoff locations. They happily accept used items, but please only donate things in good condition that you would normally pass them on to a friend.
Multiple locations: Room to Grow has locations in New York and Boston, and accepts a comprehensive list of used clothing, baby monitors, bouncy seats, blankets, etc. for babies who could use a little help. A few items — like bottles and cups — they will only accept new, for hygienic reasons.
You can also try your local battered women’s shelter or homeless shelter.