The First Minister said the company's announcement, which said it had been left with "no option" but to wind up with months, would be of "serious concern to many people across Scotland."
SYT says it willl cease trading at the end of July after being turned down for long-term funding by Creative Scotland, along with several other theatre companies.
The government stepped in, along with a private-sector backer, Clyde Blowers, to bail out the company four years ago after a previous rejection by the quango.
The closure announcement by SYT’s board has come less than three months into a multi-million pound Scottish Government campaign to promote 2018 as the Year of Young People.
Creative Scotland, led by chief executive Janet Archer, has already been forced to apologise to the Scottish Parliament last month for the handling of recent funding cuts and has pledged a “root and branch review” of the way decisions were made.
Scottish Youth Theatre had warned in January that its funding rejection by Creative Scotland had left it facing “serious challenges.”
But unlike several other companies which were reprieved by Creative Scotland after an emergency board meeting to deal with widespread criticism across the cultural sector, SYT was left empty-handed.
It has been funded to the tune of Â£400,000 by the quango for the last three years, but this has been ring-fenced support paid for the government and private-sector firm Clyde Blowers.
Ms Sturgeon insisted the government could not intervene over Creative Scotland’s decisions on who should receive "regular funding" - but pointed out that it had been able to explore other options previously when SYT had been turned down for support by the quango.
Ms Sturgeon was asked about the impending closure of SYT by Scottish Greens co-convener Patrick Harvie, who insisted the government had a “direct responsibility” to take action to secure the future of the national youth theatre company, pointing out that they even performed at the opening of the current session of the parliament.
He added: “This year is being billed as the Year of Young People. Are we really going to let Scottish Youth Theatre close?”
Ms Sturgeon said: “The announcement by Scottish Youth Theatre...is of serious concern. It will be of serious concern to many people across Scotland - and indeed it is to me.
"I know Creative Scotland has approved some funding to allow the organisation to continue to operate while they hopefully work together to look at alternative routes to support.
"I'd encourage Scottish Youth Theatre to continue these discussions. I've also asked the Culture Secretary (Fiona Hyslop) to offer to meet with them to see if there is some action the Scottish Government can be party to party to that would help to secure a future for Scottish Youth Theatre.
"Yes, we are in the Year of Young People, but more generally than that, arts - and theatre within that - are really, really important to the well-being of our country. We want to make sure that theatre and youth theatre in particular, can flourish, not just this year, but generally in Scotland."
Ms Sturgeon said Creative Scotland was sometimes "unfairly criticised" when difficult funding decisions had been made, but also pointed out that its budget had been increased by the government in its most recent settlement.
She added: "Scottish Youth Theatre do fantastic work and I think it would the desire of all us to see them able to do that.
“Scottish Youth Theatre was not previously in receipt of regular funding, as I understand it. When decisions were taken in the last round of that my predecessor as First Minister (Alex Salmond) was involved in exploring options at that time. We will continue to work with Creative Scotland and Scottish Youth Theatre.
“I have asked Fiona Hyslop to meet with them and while I cannot give detail about what those options might be, I will certainly give a commitment that we will do everything we can to fully explore all options to allow young people in the future to benefit from Scottish Youth Theatre in the way that young people in the past have done.
"We are determined to look at all options to protect if we can the work that Scottish Youth Theatre does."
Meanwhile the convener of Holyrood's culture committee has blamed Creative Scotland for triggering the impending closure of Scottish Youth Theatre.
Joan McAlpine, who branded Creative Scotland's explanation of its decision-making processes "completely unsatisfactory" after a hearing before MSPs last month, said: "Creative Scotland admitted that its process for selecting regularly funded organisations was flawed in evidence to the Scottish Parliament's culture committee two weeks ago.
"It was so flawed that they plan to completely review it. Now we learn that a decision made in that flawed process has resulted in the announced closure of Scottish Youth Theatre. This is serious. It's unclear what Creative Scotland have against SYT - they refused the organisation funding in 2014 and it was only the intervention of former First Minister Alex Salmond that kept it going."