Confluences may be scary and stressful, but they aren't the end of the world. Whether or not you're a DRT worker, there are always things you can do to be prepared for these events and keep yourself safe.
A guide specifically intended for DRT workers can be found here.
The most effective way to navigate an emergency is to make a plan beforehand. Familiarize yourself with dungeon survival best practices and make sure you know what you'll do if you or a loved one is kidnapped by a dungeon.
Before a dungeon confluence officially starts, there will be a period, typically lasting a few months, where the frequency of dungeon breaks will markedly taper off. Appearances of S-, A-, and B-rank dungeons also decrease during this period.
Confluence onsets, informally known as anti-confluences or the calm before the storm, are a good time to review your confluence preparation plans and make sure you have any extra supplies you may need. Official sources like the DRT and local governments do not typically announce confluence onsets, partly because there isn't a defined beginning point or consistent duration for onsets. Unofficial sources may make announcements, and official sources will typically circulate confluence-specific resources during suspected confluence onsets.
NOTE: Due to the nature of known precognitive esper powers, it is functionally impossible to predict the date or time a confluence will officially start. Someone who claims to have definite advance knowledge is almost certainly mistaken.
When the beginning of the confluence has been confirmed, official sources will make announcements. It can be a good idea to share these announcements with your friends and loved ones; it is particularly important during dungeon confluences to verify the reliability of news articles you share.
Confluences are disruptive to everyone, not just the DRT. You may experience delays or interruptions in non-emergency services; please exercise as much patience as you can. Everyone you meet is likely to be under as much stress as you are, if not more.
If you have the time, consider volunteering! The DRT is always accepting help; this page has a list of support roles that don't involve going into dungeons.
Dungeon survival during a confluence is different than it is normally. A list of guides for some of the named S-rank dungeons can be found here.
If you are kidnapped by a dungeon during a confluence and you are not injured or being kept in one place by your environment or dungeon monsters, start moving. C- and D-rank dungeons are much more likely than usual to prematurely close during confluences. If your phone works, put it in dungeon mode and use the DMM system to inform other kidnapees of any useful information, like routes to the exit, but don't waste time. Put on your own oxygen mask first.
Post-dungeon procedures during confluences will likely be different than otherwise. DRT workers will be operating under a heavy workload and any espers you encounter will be operating under a heavy workload while backlashed. As always, a little extra patience will go a long way.