Not on your tax return. If they earned more than $12,200, a tax return has to be completed in their name. If they earned less than $12,200, they are not required to file a return, but you should if they had any federal tax withheld, to get that money refunded.
This is the first year our son had a job. I started separate returns. One for my wife and I and one for our son. We claim our son as a dependent and then on his return noted that he is claimed by someone (us).
Is this the correct way to file?
Yes, this is exactly the correct way to file.
Dependent children with $12,200 or more of earned income in 2019 will need to file a tax return. If they earned less than $12,200 they are not required to file but may want to do so to recover any monies withheld for taxes. Dependents who are 65 and over and/or blind have a somewhat higher filing requirement.
Dependents who have unearned income (interest, capital gains, dividends) will generally have to file if the unearned income is more than $1,100 for 2019. Once again, income levels are higher if the dependent is blind or 65 and older.
A parent can elect to claim a child's unearned income on their return, providing certain criteria are met. Beginning in 2018, all net unearned income over the threshold is taxed using the tax rates for trusts and estates. The threshold for 2019 is $2,200.
You can find more information here and here.