It shouldn’t. Most home security systems that aren’t hardwired communicate through cellular or radio signals. The Wi-Fi may be there as an aid, but it’s not the end-all-be-all. The other communications still work when the Wi-Fi is down. The exception is if the system clearly states that it works through the internet only. If the system is solely VoIP, it may stop working when the Wi-Fi goes down. One final note: if the power goes out, the system will last only as long as the backup battery. Ask your security provider whether you’ll get a notification when your equipment shifts to the backup battery. You’ll want to work quickly to get things back online. Otherwise, your system will stop working when it runs out of juice.