Whether someone pays federal income tax is based upon the person's income, regardless of age.
A single person under 65 years of age who earned more than $12,200 in 2019 was required to file an income tax return, but at that level was unlikely to owe any income taxes if eligible for the standard deduction of $12,200. (The standard deduction for 2020 was raised to $12,400.) At the federal rate of $7.25, which applies in 21 states, a minimum wage-earner would have had to work 1,683 hours in the year before crossing the filing threshold. Minimum wages vary considerably across U.S. states and cities, ranging up to $15.59 in San Francisco.
In addition, workers pay social security and Medicare taxes from the first dollar of earnings.