There is no one objective answer to this question. The answer depends greatly on what things a person thinks are most important and on the level of trust that a person puts in technology.
On one level, this is a question about how much we trust the technology used to extract and transport oil. If we believe that modern technology is good enough that drilling can be done with only a very few accidents, then we will have fewer objections to drilling. By contrast, if we think that accidents are more likely to occur, we will be less likely to approve of drilling.
However, the major issue in this controversy is what we value. There are two main competing interests in this issue. On the one hand, there is the health of the environment. On the other hand, there are economic issues. Drilling for oil brings economic benefits. It brings jobs to an area. It reduces the price of oil because it creates a larger supply of oil. It can reduce the degree to which the US must import oil from other countries. All of these are economically beneficial. (It should be mentioned that there are those, like fishermen and resort operators, who worry about the economic damages that could be caused to them by oil drilling and spills.)
However, there are some who think that environmental consequences outweigh economic gains. These people point to things like the environmental devastation of the Exxon Valdez spill in Alaska in 1989 or the Deepwater Horizon spill in the Gulf of Mexico in 2010. They also point to the role of fossil fuels in creating global warming. They argue that the environmental damages from large spills (and from the use of fossil fuel in general) are much greater than the benefits that can be gained by drilling.
This issue, then, is really an issue of values. There is no objectively right answer because there is no objective way to determine whether the environmental impacts of such drilling outweigh the economic benefits.