Domain B Trusted by Domain A? If so then you shouldn't need to do anything.
Just type the name into the people picker in the form of Domain\username and
it should resolve. If there isn't a trust relationship then you would need
to extend the web app to a new zone and implement an ADMembershipProvider
that points to Domain B. Or create a trust between Domain A and Domain B.

We have been looking at how to do this in an extranet where there is no trust
from the DMZ domain into the internal AD. We have been looking at ADFS but there
are various issues with things like the people picker and the amount of manual
administration.

Someone suggested the product from http://www.optimalidm.com which is a
directory proxy. It can present multiple backend directories to SharePoint as if
they were one LDAP directory. It can create dynamic virtual groups based on
directory attributes. They also have a custom people picker, although it looks
like only for SP2010.

Does anyone have any experience with this product they can share?

If this is the case, wouldn't it be a matter of the DMZ domain trusting the
internal domain? Is the concern about exposing internal log-in information to
the DMZ network?

In this environment, security do not want to allow a trust to the internal
network.