iPhone 4s is not responding at all
It could be the battery.  The phone is 3-4 years old.  However it could also be the charging cable or the port.  The battery symbol with a sliver of red just means the battery has no charge.

If you can, try using other charging cables to see if those are more successful in charging the phone.  

If different cables don't work, then try cleaning the contacts in the charging port.  Those can get dirty or corrode over time.  Try some of the tips here to clean the port: http://www.cnet.com/how-to/use-a-toothpick-to-clean-out-your-iphone/

If none of those work, you should look to try to replace the battery.  Apple can do this and there are 3rd party services like iFix that will charge $40 or so to replace the battery.