Is it a bad idea to give an infant (1 year old) antibiotics for a cold? I feel like a cold is a minor thing and may be good to build immunity.

Yes. If the baby truly has a viral illness then antibiotics are unnecessary and also potentially harmful in many ways, not the least of which being thinning of your wallet.
Dont us antibiotics. The common cold is caused by a virus. Antibiotics have no activity against viruses. There is no reason, and there may be some harm, in using antibiotics for viral infections.