I married last year and my now wife has two children, when I file jointly this year do I count my step-children as dependents? As my child or another person?

Yes, you can add stepchildren as your children to your joint tax return if

1. Nobody else is claiming them.

2. The children must be under age 19 or, if a full-time student, under age 24. There is no age limit if your child is permanently and totally disabled.

3. The children must have lived with you for more than half the year, but several exceptions apply.

4. You must provide more than half the support.

Click on the link for more information

Dependents

@brian101ochoa 

On your tax return, stepchildren are treated the same as if they were your own children.

Furthermore, on a joint tax return dependents are not designated as dependents of one spouse or the other. They are dependents of the couple filing the joint return.

When you enter your wife's children in your joint tax return, you are filing for both you and your wife, not just for yourself. You enter them as your son or your daughter.