Get contacts from broken iPhone 4S
If your phone is linked to an iCloud account, try surfing to https://www.icloud.com and login with your credentials.  Once logged in there is a Contacts app which should contain your address book.

If you don't have iCloud you could use iTunes as stated above, but it won't let you access your contacts directly.  Try alternative applications such as Syncios, Appandora or iTools.

If that's not an option you could try to backup your old iPhone with iTunes and restore it to another iPhone.

Another option is to replace the screen and hope for the best.  While repairing a 4S might not make sense, there's cheap refurbished or OEM screens available.  If you don't want to use it afterwards, it till makes a great gift for any kid.  Replacing a screen on a newer iPhone (5 and above) is dirt easy, but replacing one on a 4 or 4S is absolutely not for the faint of heart.

If none of the above work out for you I'm afraid it's a lost case.