A nephew will visit me in Phoenix from the Ukraine. A travel agent claims he needs to take an amount equal to $50/day for each day he's here. True?

Your nephew needs to establish he will become a public charge. This is typically done by having sufficient funds available to cover the expenses of a trip to the United States. There is no set number amount required, but if your nephew cannot establish how he will be able to stay in the United States he could be denied entry. You can read more about visitor visas at http://myattorneyusa.com/travel-visas.