For your CV, I would list my positions as:
Postdoctoral Fellow [or similar and official title]
Broad Institute of MIT and Harvard. Cambridge, MA.
2023-Present
Thier webpage uses this order. You're at both and probably neither. That is, your institution is a joint venture, so it is affiliated with both, but you are not. You would work for the joint venture. Unless you have a second appointment with a department at MIT or Harvard, I would not list those universities.
I suspect Broadies have access to both schools' libraries or their own subscriptions to online resources.
As for other support, it looks like their post docs have their own support programs:
Each selected postdoctoral researcher will be provided with a stipend, travel allowances to attend scientific conferences and meetings, and retirement and health benefits provided by the Broad Institute. Funding is provided for up to three years. In addition, participants will receive mentoring and professional development opportunities.
I would encourage you to poke around their website, and, if you're serious about postdoc position there, reach out to PIs and current postdocs who can tell you what it is like to work there.
Lastly, in the grand scheme of things, I suspect listing the "Broad Institute" is more important than listing yourself as being affiliated with Harvard and MIT. They have their own branding. Additionally, your own productivity (e.g., publications, grants, patents) will be more important than where you accomplish them for the next step of your career.