How do antibiotics affect protein synthesis?

Generally not: antibiotics are essentially antibacterials, that is drugs counteracting bacterial infections like pneumonia, salmonella and the like.

The specific mechanism of working of antibiotics depends on the specific drug under examination.

Penicillin and its derivatives are bacterial killers: they enters into the bacteria membrane and inhibit basic bacteria life processes. A few of these processes are also related to bacteria proteins syntheses and, under this point of view, these class of antibiotics are also protein synthesis inhibitors. The important think to remember is that they inhibit, among other effects, the synthesis of bacterial proteins, not of the host (generally a human) protein syntheses.

Other antibiotics inhibits bacteria multiplication by thwarting bacteria DNA splitting and replication.

Also in this case, since many proteins are involved in the DNA replication process, bacteria proteins are involved, but the main antibiotics effect is to inhibit DNA replication.