Flavius
"Mettle" and "metal" were alternative spellings for a same word, which was a favorite of Shakespeare's for making puns. One meaning, for which today we use the "mettle" spelling, is a person's character. Flavius is wondering if the base, or low-class, character of the plebeians has been affected by his and Marullus' scolding.
The second meaning, today indicated with the spelling "metal," refers to hard, shiny, malleable materials. Common metals such as copper or tin are referred to as base metals, as opposed to precious metals such as gold and silver. Back in Shakespeare's time, alchemists claimed to be able to change base metals into precious ones, which explains the wordplay in Flavius' phrase, "See whether their basest mettle be not moved."