It is definitely a complex problem with no simple solution.
But consider this: if open borders could be the solution to the elimination of poverty, why is there poverty in the US, the UK, Germany......among the "native" population? If it were solely a matter of having access to opportunities, then it would seem that all those with access would be living the good life. But sadly, that isn't the case, and again part of that is due to differences in individual abilities.
Poverty is universal and the reasons are complex, and I doubt whether there IS a solution barring a drastic reduction in the population completing for finite resources. When I took my first population geography class as an undergraduate back in the 70's we talked about this, and discussed the concept of Zero Population Growth, which I personally support.
In 1970, the estimated world population was 3.7 billion; currently it's move than 7 billion. Even in the 70's, there was real concern that the earth could NOT sustain the 3.7 billion never mind 7+ billion we have now. Poverty has always existed in relative and real terms, and even with a decline in the rate of growth, will continue to exist. Eliminating it for an individual via access to opportunities in Country A does not eliminate poverty for the residents of Country B.
It's not a selfish position; I have no issues with anyone who wants a better life whether at home or in a new country. I just believe that open borders won't solve this particular problem, and agree with PlainPearl that it's an overly simple solution for a complex issue.