You don't take a "standard deduction" for your dependents. You may be confused with the old rules for dependents (although the terminology still isn't quite right).
First, if you and your spouse are both US persons, your best tax result will be to file jointly. A "US person" for income taxes means you are a citizen, a green card holder, or someone who passes the substantial presence test--meaning you have lived a certain number of days in the US, regardless of your authorization to be in the US or to work.
https://www.irs.gov/individuals/international-taxpayers/substantial-presence-test
If you are both US persons, which deduction you use depends only on your filing status (married filing jointly or married filing separately). Married filing jointly can be either itemized or standard deduction; if you file separate returns under married filing separately, both returns must make the same choice.
Then as to dependents,
Any dependent who is a US person will qualify you for at least a $500 tax credit. To qualify for the $2000 child tax credit (and to qualify for Earned Income Credit) the child must have an SSN that is valid for work. The IRS says "For the EITC, we accept a Social Security number on a Social Security card that has the words, "Valid for work with DHS authorization," on it." So you don't necessarily need the actual authorization, but the children at least need to be eligible for authorization.
If you and your spouse are Not "US persons" for income tax (because of a Visa or other rules) then you need to follow up with @pk
