As I understand it, the father can claim all of his kids and provide them medical benefits from his job. It's no different than a man with kids from previous wives or girlfriends that he's financially responsible for. That's why on Sister Wives, Kody and Meri got divorced. Kody was then able to adopt Robyn's kids from her first marriage and provide them with his medical benefits.
Otherwise, only the legal wife can apply with the husband for loans, file joint taxes, etc. When the Sister Wives family purchased the four homes in Las Vegas, they said that Meri (who was the legal wife at the time) was able to file jointly with Kody for her mortgage and have his income considered as well as hers. The other three wives had to file as single women. It seems that the Brown family keeps their finances somewhat inter-mingled. Kody has a job, as does Janelle. Christine got her real estate license several years back, although it's not clear whether she's actively working in the field. The family has a business that sells jewelry and clothing. Robyn seemed to be most involved in that. Otherwise, they get income from the doing the show (which I read has dropped considerably recently as ratings have declined). Meri had to approach the family for approval to buy the bed and breakfast she wanted.
I've read books about LDS communities that are much more strict and seem almost cult-like. The man might have six or seven wives and over 30 kids. In those communities, many of the wives do file for government assistance. Some of the wives have jobs while the others take care of all the kids. The ones who work must hand their paychecks over to the husband. He gives out the money to the family members as he sees fit. He also controls the money of those who get government assistance.
On Seeking Sister Wives, they never really discussed their finances. The Briny dad is an attorney. He probably makes good money and provides benefits to his kids. No idea what Dmitri does for a living (besides womanizing!).