As Arkcon said, all life that we know of is carbon based. This is a red herring. The carbon-based nature of many compounds used as drugs has nothing to do with the way that they affect bacteria or with the ways bacteria develop resistance to them. In fact, bacteria develop resistance to silver as well - see for example
http://www.sciencedirect.com/science/article/pii/0147619X9290008XThe trick, as Babcock-Hall pointed out, is to find a process that bacteria require to survive and find a way to block that process, killing the bacteria. We can't use a process that we also need to survive, or we will kill ourselves as well. Either that, or we need to make sure that whatever compound we are using to kill the bacteria never reach portions of our own bodies where they will be toxic.
One of the differences between mammalian cells and bacterial cells is that bacterial cells use different respiratory pathways which involve reactions on sulfur-containing proteins. Silver ions bind especially well to sulfur, and are easily taken up into bacterial cells. Once inside, they tie up the sulfur atoms preventing respiration (among other things) and the bacteria dies. Mammalian cells have different respiratory pathways and are less reliant on oxidation changes in sulfur atoms. So silver hits bacterial cells much harder than mammalian cells, and by carefully selecting a dose, you can kill lots of bacteria without killing too many mammalian cells and getting undesired side effects. Also, human skin is pretty good at blocking the entrance of ionic compounds, so if the bacteria is on the surface of the skin, you can use a much larger dose without causing too much damage.
Bacterial cells could evolve resistance to silver in several ways. One would be to become less reliant on the sulfur mediated respiration systems and substitute other pathways, one would be to sequester the silver atoms inside the cell in vacuoles or in scavenger proteins, one would be to evolve efflux pathways that send any silver ions back out of the cells as they enter, and one would be to develop membranes that are less permeable to the entrance of silver. All of these pathways seem to be followed to some extent in different types of bacteria.
Bacteria is actually a fairly easy target, in one sense. Bacteria cells are very different from mammalian cells, and there are a lot of different mechanisms that can be targeted in bacteria without doing too much damage to the mammalian cells they might be associated with. However, many bacteria have evolved effective mechanisms for changing their genomes and protein expression, which allows them a lot of diversity when faced with a biological threat. Resistant strains develop fairly rapidly to many challenges.
As the cells get closer and closer to normal mammalian cells, it is harder and harder to kill undesired cells without causing serious side effects. This is one reason that cancer is such a difficult target - cancer cells are human cells, using for the most part normal human processes for growth and division. The only options for developing cancer treatments are to find the very subtle differences and target those processes alone, or to find ways to keep the drug localized at the tumor without being distributed through the rest of the body. This is also one reason that viruses are such difficult targets. The viruses aren't truly alive and thus can't be killed by interfering with their life processes like bacteria can. They don't become active until they have actually infected a human cell, and now you have to find a way to kill that cell without killing the uninfected human cells. Fortunately viruses can be recognized as non-human by the immune system and targeted for destruction, while cancer cells having been human all too often are not targeted by the immune system.
Just wandering - I hope some of this is useful.