Per Delta Sky Club House Rules on its website, no guests are permitted for one-day pass. But you can always check with the sky club when you are there, they may let your kid in. If they do, buy the pass there. If they don't let your kid in, don't buy the pass.

One-day Pass
•The cost of a One-day pass may be credited within 60 days of the purchase date toward the cost of an annual or three year* membership.
•Individuals must be at least 18 years of age to purchase a one-day pass, and 21 years of age to access Clubs with a self-service bar, unless accompanied by a parent or legal guardian who is also accessing the club with a one-day pass or who is a club member.
•No guests are permitted with a One-day pass.
•A One-day pass must be used in conjunction with same-day ticketed air travel but such travel may be on any airline.
•One-day passes are valid for a single day of travel. Itineraries that include a date change between origin and destination are treated as the same day of travel.
•One-day pass is valid for use at any Delta Sky Club location. This pass does not grant access to partner lounges.
•One-day pass purchased online, is valid for one year from the date of purchase.