What are the benefits of genetically modified organisms, as in corn, soy, wheat, or rice?

Actually most of the food you buy is not genetically modified.  It is not required by law to label foodstuffs in the US, but most food products are not using GM varieties.  You asked the benefits of the GM foods, in most cases there are no advantages (or disadvantages) to the food itself, with the exception that there may be more food available because GM varieties were used. Most of the advantage of GM crops are in the growing the crop itself.  You would need to take them on a crop by crop basis to see the advantages resulting from GM crops.  The biggest advantage can be seen in weed free soybeans today compared to several years ago when weeds almost overpowered the crop.  It requires much less herbicide to control the weeds in soybeans today than before the GM varieties were developed.  I know that this the opposite to the information being drilled into you by the activist groups opposed to GM crops.  They say that the plants keep requiring more and more pesticides  and more pesticides will be showing up in the food we eat, but this is just not true.  There are fewer advantages to GM corn and less of it is being grown than soybeans.  Wheat, as far as I am aware there have been no GM varieties being grown at this time. Now about the rice, This is the exception to the rest of the crops, where the advantage is to the food itself.  Golden rice has been developed which has large amounts of Vitamin A precursor. It also contains larger amounts of iron. This is something lacking in regular rice and when rice is used as the primary food source, as it is in a lot of countries, leads to a Vitamin A shortage and many health problems, including blindness in children, and death if the deficiency is not corrected. Greenpeace and other activists organizations have fought the release of Golden Rice delaying it's use for years while the blindness and death have continued.  They will use half-truths and outright lies as propaganda to keep  the GM crops from the people that need them.  
I think that people have the right to know what they are eating and if they choose to avoid GM foods they should have the right to do so.  GM foods will have to be labeled to assure this.  Your choice should be based on scientific facts and not politics and activise properganda, however.  As you stated about the only way now is to choose organically grown food, even there you can't be sure unless you grow it yourself or know the grower.

As the above answer stated, actually most of the food is not GM.

The largest GM crops are soybeans, cotton and corn. (of course you could argue that since ALOT of commercial foods contain soybeans and/or corn that in fact most food is GM)

The benefits are many.
Plants that carry human vaccines
Plants that are vitamin enriched
Plants that are insect resistant
Plants that have a higher yield
Plants that are disease resistant
Plants that grow quicker
Plants that are more resistant to heat/drought, cold/frost
Plants that have increased nutrients
and probably many more, but those I have all read about in current production

The drawbacks is what is not so easy to see
It really is unknown what the long term effects of growing and consuming these crops will produce.  Scientists do not yet have the knowledge to fully understand the long term impact of these plants.  There is plenty of proof that they will cross with non GM crops and even wild plants.  One type of GM crop, modified to have natural insecticides, has been proven to be the cause of some bee colony kills and GM crops are being looked at more closely in the bee colony devastation currently going on.  I doubt that they will be the overall cause but they might contribute.

I have read that the genetically modified food can made people not very reproductive(fact is that in earth have 6 bilion people)