Is there a Debian-based distro with only the core packages(like apt, dpkg, ifupdown, nano, ssh, grep, sudo, coreutils, wget, etc., and no X11)? And not meant to used on servers and it must be 32-bit. It must be available on USB (.iso file). I only need a basic Linux system and I don't have much space on my (4 GB) SSD. I know how to manually configure ifupdown. My laptop is an Asus EEEPC 701
I can recommend the Debian NetInstall.
The goal of NetInstall is to have a minimal working linux up and running. The idea being that you then install the packages that you are missing manually.
Keep in mind that all the packages you listed are very small to begin with, so installing anything that NetInstall might be lacking is easy (given that you have a ethernet connection, I never tried using WiFi from the beginning and doubt that it works).
If you don't have a network connection you have a second way of obtaining the packages you need, that might not be shipped with NetInstall. That is: Download one of the larger Debian ISOs and burn it to CD/DVD. When you need to install something, insert the CD and point your apt sources to the disk. That way you are flexible but don't need any network connection.
An alternative for this would be to get the large Debian image directly and go through the menu installer manually to select only the parts that you want. (That is: Don't select the X11 etc. elements). That requires some more time on your part but works well.
After that you have the large disk as a resource to install everything that you might need without network connection.