The CBD is not itself a binding law, and it does not have any enforcement mechanisms. What it does is place responsibility for regulating access to genetic resources in the hands of the nations providing those resources. So, aside from the fact that the US is not a party to the CBD, the CBD itself cannot require any US public garden to do anything.
ABS is turning out to be complicated (to say the least) and is still a work in progress. Individual nations have created their own laws governing access to genetic resources, and those nations have the right to enforce their laws in whatever way they have chosen. Some nations have a strong national authority (e.g., India) and give little say to local communities. Others, such as the Philippines, have given more say to indigenous peoples. Costa Rica requires consent from both local interests and a national authority. Few African nations have created ABS laws - Ethiopia didn't create a law until 2006. Critics of these things have suggested that some developing nations' ABS laws are not well-thought-out, and researchers have complained that ABS procedures can be so idiosyncratic and onerous as to make field collection hardly worth the trouble.
One problem that some scientists have identified with many national ABS regimes is that the laws seem focused on incredibly lucrative bioprospecting that results in multi-gazillion dollar drugs - not on, say, modest seed exchanges between gardens in two different nations. The Nagoya Protocol, which was just added to the CBD and hasn't exactly been ratified, is intended to solve some of these problems by simplifying access to genetic resources for non-commercial use and by creating an ABS clearing-house to share ABS information among countries. This protocol also encourages the sharing of technology, which could be one of the "benefits" in exchange for access.
So to sum up this quick note: ABS is different for every nation. A garden that wants to work with a particular nation will have to follow that nation's ABS laws, which will involve finding out what those laws are, figuring out which national/regional/local authorities need to grant permits, and possibly negotiating an agreement with the local area where the genetic resources are found. There is no standardization.
Within the U.S., individual gardens have started adopting their own ABS provisions, in a sense binding themselves to the CBD as institutions. As a practical matter, any U.S. garden that collaborates with gardens in other nations is effectively part of the CBD community. Building trust and sharing information and resources are essential to global plant conservation anyway - selfishness is counterproductive.