SAMHSA (The Substance Abuse and Mental Health Services Administration), an agency in the US Department of Health and Human Services, provides resources for individuals and organizations to learn about and deal with trauma. The agency's Community Trauma Initiative defines an organization or community as trauma-informed if it:
REALIZES how widespread trauma is
RECOGNIZES the signs and symptoms of trauma
RESPONDS by integrating knowledge into practice
RESISTS doing further harm
Graphic from The Institute of Trauma and Trauma-Informed Care, University at Buffalo
Social services and health organizations and agencies have been at the forefront of trauma-informed initiatives. But libraries are also in a position to support these initiatives while embracing their practices.
The team behind Philadelphia's Healthy Library Initiative note that "libraries are disproportionately frequented by vulnerable populations, including those experiencing mental illness, substance use, and homelessness." 
Furthermore, a 2014 case study explored the public library's role as a therapeutic landscape, concluding that
[T]hree aspects of the public library as a therapeutic space were important; the familiar and welcoming environment, quiet, calm atmosphere, the empowerment associated with being able to make non-commercial, unpressured decisions about what to read all contributed to the opportunity to conduct an act of self-care by withdrawing from stressful situations into the public library space. [...] These previously hidden benefits show that the public library may help with the everyday recovery work of mental health, providing a sanctuary in the face of stress. 
In offering a potentially therapeutic space to their patrons, especially those who are vulnerable and affected by trauma, as well as an ongoing slate of educational, recreational, and community-building programs and services, libraries will recognize their natural role in promoting a trauma-informed approach.
 Morgan, A. U., Dupuis, R., D’Alonzo, B., Johnson, A., Graves, A., Brooks, K. L., … Cannuscio, C. C. (2016). Beyond Books: Public Libraries As Partners For Population Health. Health Affairs, 35(11), 2030–2036. doi: 10.1377/hlthaff.2016.0724
 Brewster, L. (2014). The public library as therapeutic landscape: A qualitative case study. Health & Place, 26, 94-99. doi: 10.1016/j.healthplace.2013.12.015