From the point of view of an environmentalist, which is probably the most significant argument against offshore drilling for oil?

Since I didn't know how best to answer you, just for grins I Googled keywords from your question and got a bunch of hits. These sites offer lots of reasons offshore drilling is not an environmentally sound idea:

HowStuffWorks "Effects of Offshore Drilling: Energy vs. Environment" (http://science.howstuffworks.com/environmental/energy/offshore-drilling-controversy2.htm)

Worth the risk? Debate on offshore drilling heats up - USATODAY.com (http://www.usatoday.com/money/industries/energy/2008-07-13-offshore-drilling_N.htm)

You will have to decide on the most significant reason and be able to support your decision.