Dependents who are in the U.S. and who need to make a change of status or extend their stay concurrently with the change or extension of the principal visa holder need to complete the Form I-539. This is a CIS form, which can be found and filled out at http://www.uscis.gov/i-539 -- it is strongly recommended to fill the form out on-screen, for reasons of legibility.

Note the following instructions when filling out the Form I-539:

The name of the "first dependent" (usually the spouse, but can be an elder child if the family structure is such) is intended for Part 1. The supplement is used for subsequent dependents. DO NOT USE PART 1 FOR THE PRINCIPAL APPLICANT'S NAME.
In Part 1, question 12, the dependent will indicate the expiration date of their current status. Dependents currently in F or J status will select box 12.c.
In Part 2, item 2.a., if the dependent is changing status, the effective date of change must conform to the start date of the principal's H-1 petition; this may have to be verified with the sponsoring department, since it depends on funding and other variables. Ultimately it will appear on the I-129 prepared by Bechtel, which is the operative document for these matters.
In Part 2, item 5.b., the total comprises all the people mentioned on the form, including the supplement, if it is completed. Don't include the principal.
In Part 3, item 1.a., the date must conform to the end date of the principal's H-1 petition; this may have to be verified with the sponsoring department, since it depends on funding and other variables. Ultimately it will appear on the I-129 prepared by Bechtel, which is the operative document for these matters.
In Part 3 item 2.a., "no" is almost always the correct box to check. An affirmative response would be appropriate only if a spouse were changing from an independent classification and the principal had previously secured an extension or change of status.
In Part 3, item 3.a., the response is almost always "yes, filed with this I-539." "No" applies to individuals who are requesting an independent change or status or extension of stay.
In Part 4, the information requested pertains to the individual named in Part 1.
In Part 4, question 19, the dependent must indicate his or her means of support. Many spouses and children have no employment history, and all dependents who are changing to H-4 status lack employment eligibility. The detailed description will be entered on page 6 in section "Part 4. Additional Information for Answers to Item Numbers 18., 19., and 20".
- If the dependent's history does include employment and answers "yes", develop a statement to describe the name of the dependent who was working, the name(s) and address(es) of employer(s) who paid the dependent, and the income. A copy of the EAD or other authorization must be included among the immigration documentation (see below).
- If the dependent(s) have no employment history and answers "no", a statement like the following will be sufficient: "The person (or persons, if applicable) whose name(s) appear/s on this form are supported by the principal applicant (name of principal), who is the beneficiary of the related petition filed on his/her behalf by Stanford University with an annual salary of ___."

More information

Dependents of H-1B employees who have entered the US in H-4 status or who have changed to that status from another non-immigrant status may remain in the US until the end date noted on their last I-94 card. If the H-1B employee extends his or her H status, the dependents must also extend their status by concurrently submitting a request for an extension.

Dependent H-4 students are permitted to study in the U.S. but they may not work. H-4 status expires for children at the time they reach 21 years old.

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