Usually you have to give the details about the members of your household, dependant or non-dependant. This is applicable in bankruptcy, an Individual Voluntary Arrangement, and debt management, the major debt solutions. This is because the creditors usually require a copy of your household income and expenditure, and the only way to back up what you spend or bring in is to verify the details of the members of your household.
Regarding your children or dependants it is likely you will only be asked for their names and dates of births, literally for proof they exist. In terms of your income and expenditure, this would back up and tax credits you get, any child benefit and also outgoings ranging from clothing and food through to nursery, child minding and school expenses.
Giving this information is in no way going to affect your child as they are usually to young to actually have a credit report, and even if they did it wouldn’t be registered on there anyway as the law changed several years ago regarding what is registered on your credit report. It is no longer the property or family name that carries the credit report, it is the individual it is recorded against. If your children or any one else for that matter is connected or affiliated to you by means of your credit report by accident or by means of a previous financial connection (e.g. a mortgage or loan), you can write to the credit report companies such as Experian or Equifax and get you or them disassociated from them or your respectively.