There are two reasons I've seen for objecting to GMO, one unreasonable and one reasonable.
The unreasonable objection, which is probably the most common, is based on the belief that there is something inherently dangerous about GMO.
The reasonable objection is based on the belief that there is nothing wrong with GMO itself since it is just another tool and so whether it is good or bad in a particular instance depends on the user rather than the tool, but many (most?) of the current users do not have the inclination and/or wisdom to only use it for good.
Many of the interests of food producers (I'm including everyone in the chain before the food actually reaches the eater) are not aligned with the interests of food eaters. For instance a producer of a vegetable might prioritize uniformity of size and shape and predictable growing rates over taste and nutrition, because the former directly affect how effectively he can automate production. As an eater, I care a lot more about nutrition and taste than I care about having all of my vegetables be the same size and shape.
We've seen what happens when food manufacturer interests get too far out of alignment with what is healthy and nutritious--just look at all the health issues we have in the US that are reasonably attributable to our poor national eating habits.
> GMO seems like a great idea -- modern human medicine seems to include techniques that are similar to or just-short-of "GMO."
Medicine is highly regulated. You can't just make a new medicine and start selling it. Not so with food.
For a worst case scenario from an eater point of view, suppose you've got organism X and organism Y. These are very different, and cannot be crossbred. Suppose that Y produces something that has health benefits, but that some people are allergic to. X does not produce this substance.
A person allergic to that substance has a simple strategy for staying safe: do not eat Y.
With GMO, in theory some X manufacturer could modify X to include the genes from Y that produce that substance. For most people this would be a benefit--they are now getting that healthy substance from Y when they consume X.
For our person allergic to that substance, this sucks.
Now our allergic eater has to worry about both X and Y. And since GMO foods do not have to be labeled currently, he cannot simply avoid GMO X. He now has to avoid all foods with X. (And if our allergic eater is particularly unlucky, the maker of GMO X does not announce they are doing this, and so the allergic eater first finds out about it the hard way).
Manufacturers like to tout health benefits. For our poor allergic eater, those genes from Y for that substance he is allergic to might end up being put in all kinds of foods.
My opinion is that we should go ahead with GMO, but not blindly. As a first step, it needs to be labeled, and longer term we need to develop some regulations to make sure it is used safely and wisely.