There are many clues in The Watsons Go to Birmingham--1963 that show that the Watsons are not wealthy. At the beginning of the book, the entire family is shivering in the cold because their furnace is not working properly. The furnace sounds "like it [is] about to blow up." As a result, the entire family packs into their outdated 1948 Plymouth, called the Brown Bomber. A family that had more financial means might be able to buy a newer car, as the story takes place in 1963.
In addition, it's clear that they have to be very careful about not misplacing their clothing. Their mother buys them two pairs of gloves each winter, and if they lose the first pair, they have to wear the second pair pinned to their coats so that they won't misplace their gloves. The family also has to sign for their groceries at Mr. Mitchell's store in Chapter 6. Although Momma tells the children that they are no longer on welfare (though they had been in the past), she says that they can only pay for their groceries on payday. In other words, they don't have a lot of extra money at their disposal.
Even as they head south to Birmingham, it is clear that money is a worry to the family. For example, in Chapter 8, the father buys a new record player, but the mother is very worried about the expense. All of these clues show that the Watsons struggle financially.