Your wife and the employer need to decide in advance whether she will be treated as a household employee or an independent contractor so there will be no misunderstandings later. This is part of her price negotiation too as the tax consequences are different. If the family doesn't want to deal with the details now, you are both setting yourself for hurt feelings and possibly tax penalties down the road.
As a household employee, the family is responsible for issuing a W-2 at the end of the year and for withholding social security tax. This could happen several ways, all of which are legal. Let's say the gross salary is $100 per week. The family could report a salary of $100 to the IRS, which would require them to send $15 per week in household employee taxes (15.3%). This could be made up by (1) paying your wife $100 and the family pay the $15 in tax, or (2) the family could pay your wife $85 and send the $15 to the IRS, or (3) the family could pay your wife $92.50 and send $15 to the IRS, making the family's total out of pocket either $100, $108, or $115. The case where the family pays $92.50 to your wife is most similar to a regular W-2 job. Obviously in the situation where the family only pays your wife $85, her gross really isn't $100, it's more like $92. Any one of these three arrangements is acceptable to the IRS but your wife needs to work it out in advance. The family does not have to withhold federal or state income tax (they could, but its'a lot more paperwork for them) so your wife will have to make estimated payments for the income tax, but not the social security tax.
Or your wife can be an independent contractor. The family pays the entire gross amount without withholding any taxes, and your wife is responsible for self-employment tax (social security) plus income taxes. If the employer is willing to treat her as a household employee only under situation (2) above, then her taxes will be the same either way. If the family is willing to treat her as a household employee under 1 or 3, then independent contractor has higher taxes unless she negotiates a higher pay rate. (She might offer to agree to be an independent contractor to save the family on taxes and paperwork in exchange for a higher pay rate, for example.) As an independent contractor, she might be able to deduct some work-related expenses that are not (or less) deductible as an employee.
If she in an independent contractor, the family is not required to issue a 1099-MISC, so your wife should keep good records of all her income (cash and checks). IF she is a household employee they must give her a W-2 by January 31, but your wife should still keep good records in case of problems,
The procedure for writing the income on line 7 with HSH would only be used if there is a disagreement at the end of 2017 on how she was to have been paid. If she was paid as an independent contractor but wanted to be paid as an employee, you would use this procedure, and an extra tax form to pay her 7.65% share of social security. This will trigger an IRS investigation of the family and a demand letter to the family that they pay the matching 7.65% of social security (which is a good way to cause hurt feelings and probably get fired). Or, if the IRS determines that, based on the facts of this particular situation, your wife really was an independent contractor, she will get a bill for the other half of the tax plus penalties and interest. So you really need to get this nailed down in advance.
I've attached some other information below on self-employment and household employee vs contractor.
https://ttlc.intuit.com/replies/5562810
https://ttlc.intuit.com/replies/5524669
