I don't know of any specific point & click feature to identify & alert on broken dashboards. You could probably write something using either Tableau's PostgreSQL database and/or the REST API to get the list of workbooks & data sources, then use the REST API to test access, and then from that generate alerts either using Tableau's native subscription/alerting features or using VizAlerts.
This is possible, actually, using one of the TS Data Sources: TS Data Connections. There's even an example sheet already built to answer this question, so you'd simply need to convert it into a useful alert:
So, pretty easy actually.