Beta sites are no longer as much "at risk" of being shut down as they were when for example Theoretical Physics SE was shut down well into Public Beta.
This answer by Ana reflects SE's stance on the issue back in 2015, and much of it still stands:
"The precedent was set early on that sites had two options; graduate,
or get shut down. This created a lot of unnecessary angst for our
smaller and mid-sized communities. Lots of folks on excellent sites
assumed that since they hadn’t cleared the graduation hurdle yet, we
might come in and turn off the lights at anytime. Not exactly great
Thanks to many devoted users, it’s grown clear that smaller SE sites
can do a great job of maintaining themselves and producing high
quality Q&A. Not every site is going to be a blockbuster success, but
our small sites are serving their own communities well. We’re proud of
you, and we want you here.
What does this mean? If there's enough moderation for a public beta
site to consistently remain free of spam, for flags to be cleared, and
for our Code of Conduct to be upheld, your site will remain open.
However, if community leaders drop off, flags sit without being
addressed, and we can’t find any suitable volunteers to step forward,
the site gets closed.
As of this post, not a single site currently active in our network is
at risk of being closed. Closing public beta sites is a rare
occurrence; we expect it to stay that way."
So to answer your question, about how you can volunteer to help your site survive: help keep the site free of spam, flags to be cleared, and CoC violations. You can help keep the site free of spam by flagging spam. If you're not a moderator you have no "direct" control of flags being cleared, but indirectly you can try to keep them engaged in the site by helping maintain a positive community atmosphere that doesn't drive them away from wanting to check in daily. When you see CoC violations, flag them, and encourage positive behavior in the site's chat and Meta. You may even consider running to be a moderator if an election comes up!
Other than those things, Public Beta sites need a lot of support with community building, meaning that it helps to advertise the site as much as you can (for example on Reddit, LinkedIn, other websites, blogs, etc., or even in your own social circles). More users asking good questions will help to keep the site relevant.