Why is my iPhone 4 with iOS 7.1.2 failing to connect to server and how can I connect?

Are you referring to Mobile/Cellular or Wi-Fi? If Cellular I would contact your carrier. If Wi-Fi try the steps here:

1) Restart you device. If a normal restart does not help, hold down these buttons at the same time for at least ten seconds, until the Apple logo appears: a) the Sleep/Wake button and the Volume down button (iPhone 7 and iPhone 7 Plus) or b) the Sleep/Wake button and the Home button (all other models)

2) Reset Network Settings: Settings > General > Reset > Reset Network Settings. Rejoin the Network again.

3) Reboot Router/Modem: Unplug from power for 2 minutes and plug again. Update Router Firmware (check manufacturer’s support website for a newer FW). Also try different bands (2.4 GHz and 5 GHz) and different bandwidths (20 MHz recommended for the 2.4 band). Channels 1, 6 or 11 are recommended for the 2.4 band.

4) Change to Google's DNS: Settings > Wi-Fi > Click on the Network, Delete all numbers under DNS and enter 8.8.8.8 or alternatively 8.8.4.4

5) Disable device prioritization on the router if this feature is available. Also disable any VPN apps and test Wi-Fi again.

6) Determine if other Wi-Fi devices on network are working well (other iOS devices, Macs, PCs).

7) Try device at another Network, i.e., neighbors’, public coffee house, etc.

8) Backup and Restore the Device using iTunes. Try restoring as New first and testing it. If ok try restoring the backup (the backup could be corrupted).

https://support.apple.com/en-us/HT201252

9) Go to an Apple store for hardware evaluation. The Wi-Fi chip or the antenna could be defective.

User Tip: https://discussions.apple.com/docs/DOC-9892

Hi

Follow the instructions here:

If you can't activate your iPhone - Apple Support