Simple answer: over $3000.
But taxes aren't simple. Having $3001 of earned income doesn't automatically get you $1000 per child.
The child tax credit (CTC) is limited to your tax liability. The CTC is a non-refundable credit and can only reduce your income tax to 0, It can not help you beyond eliminating your tax liability. But, if you have more than $3000 of earned income, some or all of it is usually given back to you thru the "Additional Child tax credit". That is, part of the CTC may be on line 43 of form 1040A instead of line 35 (lines 67 & 52 of form 1040)*. The ACTC is calculated on form 8812 and is basically 15% of your earned income over $3000.
*In the on-line version of TT, the main Form 1040, 1040A or 1040EZ can be seen through Tools, View Tax Summary, Preview My 1040 or go to Print Center and then choose to Preview. Form 8812 is not viewable in the online version of TT until you have paid for your return.