"Offshore" drilling is such an ambiguous term. The continental shelf extends anywhere from 20km to 400km off the coastline, which is the area that would be drilled. Considering the horizon is about 4-6km away from someone standing on a beach (since a beach is at sea level, there is no height advantage to make the horizon appear farther away), most beachgoers--and in fact all people living in low elevation states like the ones proposed for offshore drilling--will not even see the rigs. Thus the argument that oil rigs will be a blight on an otherwise spectacular beach scene fails.
And offshore oil rigs have a safety history 99.999 percent oil retention rate since 1975. That means only 0.001% of offshore oil has been spilled. The only reason people freak out about oil spill potentials is because one spill every few years gets blown out of proportion by the media and environmentalists. Oil drilling precautions and standards have also been tweaked and improved since 1975, so it is even less likely to have a severe oil spill in this day and age.
People think they are saving the environment or the Gulf of Mexico coastlines by protesting offshore oil drilling, but they should know that China and Cuba are already drilling in that exact location, and with far fewer regulations than American drillers would impose. Actually, it's probably smarter to discourage American offshore drilling, while allowing China to continue to ravage our natural resources. We wouldn't want to upset our future overlords by cutting off their oil supply.