Short answer. The minor's parent or guardian signs on his behalf. If no parent or guardian is available, an authorised representative of the Department of Labor signs instead, and the agreement is binding for its whole lifetime once signed that way. The employer and the apprentice sign as well.
What the law says
An apprenticeship agreement with a minor shall be signed in his behalf by his parent or guardian, if the latter is not available, by an authorized representative of the Department of Labor, and the same shall be binding during its lifetime.
Labor Code, Article 62 — Signing Of Apprenticeship Agreement. Read the full provision →
A signature for the minor, with a fallback built in
The rule is that an agreement with a minor shall be signed in his behalf by his parent or guardian, if the latter is not available, by an authorized representative of the Department of Labor. The order is not optional. The parent or guardian comes first, and the departmental representative is a substitute for their absence rather than a matter of administrative convenience. The point of the arrangement is that somebody with responsibility for the child has read what the child is committing to, and where no such person exists the state supplies one rather than letting the requirement lapse.
Why the agreement is declared binding for its lifetime
The same sentence adds that once signed in that way the agreement shall be binding during its lifetime. That clause exists because minority ordinarily affects the force of a contract, and the article is closing off the argument that an apprenticeship signed for a minor can simply be disowned later. It cuts both ways, and that is the useful part for the apprentice: the employer's obligation to train is equally fixed for the term, and cannot be treated as provisional because the person being trained is young.
The employer's side, and the ratification requirement
Every agreement must also be signed by the employer or his agent, or by an authorised representative of one of the recognised organisations or groups, and by the apprentice himself — a minor's own signature is not dispensed with just because a parent signs too. The article then requires ratification by the appropriate apprenticeship committees where such committees exist, and that a copy be furnished to both the employer and the apprentice. That last requirement is the one most often ignored in practice.
Ask for the copy, and keep it
If you are a parent signing for a child, do not leave without the copy the article says you are entitled to, and read the training terms before signing rather than after. Check that what the child will actually be doing matches the occupation named. If a dispute arises later about hours, pay or whether any training happened at all, the signed agreement is the document everything is measured against — and the family that does not hold a copy is arguing from memory against paper.