Genetically Modified crops can help increase the amount of food produced by the world. According to Supply and Demand, the prices of food would go down, giving people the money to buy stuff other than food. This would help alleviate poverty but would not solve it entirely.
To be honest the best way to solve poverty would take several steps.
(1) negotiate peace in countries like Sudan (war = poverty, and governmental cooperation is key)
(2) Provide widespread education (Not only in Africa but in the United States) This will give people the necessary training to get good jobs.
it is important to note that this would also include creating programs that prevent alcohol and drug abuse, both of which are associated with dropping out of school.
(3) create governmental institutions and opportunities for individuals (much similar to FDR's reforms during the great depression). This would help people get money and put it towards a productive cause (i.e. medical research or teaching)
Also create opportunities for businesses to grow (by giving tax breaks, etc). This not only provides jobs but gives people the opportunity to make their own businesses.
Only then can we "solve" poverty (although i don't think we can ever completely do it)
This is obviously a long term project that needs the strong support of the UN and other international organizations (i.e. the European Union). The best that we can do is donate to charities that are devoted to such a cause. This is the best one in my opinion