Hi Jeff,
Assuming you want to track failures upstream of the ASA and not the ASA's interface itself, SLA monitoring is probably your best bet. To be notified when the monitor fails, you can watch for syslog ID %ASA-6-622001. You can setup a custom logging list and have this syslog sent to an SNMP server like this:
logging enable
logging list sla-list message 622001
logging history sla-list
This will send %ASA-6-622001 messages to the SNMP server you have configured. Alternatively, you can also send it to a syslog server (logging trap sla-list) or to an email address (logging mail sla-list). Other than that, you'd have to have an internal server send pings out through the firewall and report back if they failed.
Hope that helps.
-Mike