It is true that you don't have to be poor to commit crimes. Yet the poor commit very different crimes than the rich. The poor commit crimes in a desperate attempt to get rich, while the rich commit crimes to get richer or to stay rich. Thus it takes money to commit the crimes the rich commit. I think that in addition to what others pointed out this is crucial as well.
I think a more enlightened discussion might be to look at why those who seem to be so abundantly blessed commit crimes. Look at Bernie Madoff. This man had millions (In all fairness, I don't know how much of his original millions were honestly earned or the result of ill-gotten gains.), and yet he bilked honest investors out of billions. He had no need, he was not impoverished, and yet he stole. Other crimes such as rape are not crimes of economic means.
As mentioned above, it depends on the type of crime, and the circumstances of those involved. If one just looks at the studies that show there is more crime in poverty-stricken neighborhoods, then this would imply that poor people are more likely to commit crimes, as if poor people have less morals. There is correlation, but correlation does not mean causation.
If we are speaking of violent crimes, then the correlation between poverty and crime may be there for reasons such as: desperation, unemployment, stress, lack of education, lack of opportunity, survival etc.
However, I do not believe it is the poverty alone that causes crime, it is the characteristics of the individuals, their family structure, their beliefs, self-control, personal goals, etc.
Millions of people are living in poverty, but only a small amount resort to crime. Many poverty-stricken people will work night and day just as hard as any higher income person, to try and provide an honest living for their families. The majority of people living in poverty would never resort to crime. Again, it's not simply the poverty, but the mentality, experiences, and values of the individuals.
Now higher income businesses and neighborhoods are also filled with crime, but different types of crimes on different scales. The kind of crime that is not as "sensational" as the more violent crimes, and also it's crime that may go un-prosecuted as it is more organized and perhaps harder and more time consuming to prove, than say a shooting. Examples: Extortion, bribery, insurance fraud, business scams, embezzlement, identity theft, theft, employee discrimination, tax evasion, money laundering, forgery etc.
As for people living in poverty who choose to commit violent crimes, giving them money would not magically change their character. Some would still find a way to commit some type of crime because that is who they have become.
I think it depends on the type of crime whether it is directly related to poverty or not. As mentioned above there are many people out there who are under stress and trying to put food on the table, for some the only way to do so is through crime. It could be shoplifting to put a meal on the table tonight or it could be involvement in the manufacture and selling of drugs as a way of making a living.
There are actually some pretty clear statistical ties between crime and poverty. Poorer neighborhoods, cities and regions consistently have higher overall crime rates, including violent and property crime. Hard to imagine there isn't a pretty concrete link.
Poverty is part of a much larger cycle of unemployment, stress on the family unit, unsupervised time for youth, exposure to violence and crime committed by others, typically less effective schools and a lifetime cycle of hopelessness that makes crime seem more attractive or even a necessity.
And since some crime requires sociopathy on the part of the criminal, poverty stricken neighborhoods usually have a more fertile psychological environment for sociopathy to grow.
It is not clear that poverty truly is the main contributor to crime rates. However, one might make the following argument to say that most crimes are, at least to some extent, caused by poverty.
One could argue, for example, that poverty puts a great deal of stress on people. They often lack material things that they need or want. They might also feel the pressure of being poor in a society that values material possessions and wealth. Both of these might lead to crime.
The desire for material goods might well lead to property crimes. People might steal in order to get money to buy the things they want or need. Or they might simply steal those things directly.
The pressure of feeling like a failure might also lead to crime. It might lead to drinking or drug use, both of which might lead to violent crime. It might lead to feelings of anger which might manifest themselves in the abuse of family members. It might also lead people to join gangs in an attempt to feel valued.
None of this is to say that poverty directly causes crime. However, it is likely that these kinds of pressures make the poor more likely to commit crimes.