This is not the popular opinion and I'm interested in facts that may dispute it. By any objective measure the problems of Haiti and the Dominican Republic differ more in severity than anything else. In other words, the social and economic problems of Haiti are more intense versions of those of the Dominican Republic, more of the same. No, the Dominican Republic can't independently resolve Haiti's problems but it has a unique role to play due to its geographic location.
Attempts to deny that unique role are irrational and illogical. There is an old saying in the southern United States; "a hit dog will holler". It means forceful rejection of criticism is a tell that the criticized party knows the criticism is truthful. I think that is what is behind the disproportionately emottional dialogue in media and elsewhere among Dominicans when the ills of Haiti and the Dominican Republic are conflated.
The Dominican Republic, like any other country, is motivated to avoid foreign or domestic policies that may exacerbate its socio-economic problems. I think the idea that the Dominican Republic does not have the wherewithal to independently resolve all the problems of Haiti and the Dominican Republic is reasonable. If foreign aid is any indication, the Dominican Republic does not have the wherewithal to resolve all its own problems. Foreign aid and foreign relations come with conditions. The wealthiest countries are not dependent on foreign and therefore determine the conditions of providing it. When the Dominican government balks at being independently responsible for Haiti, my opinion is that it is a reference to sharing its foreign aid with Haiti. In the early 2000s Jacques Chirac, the late French president caught a bit of flak when he referred to Haiti and the Dominican Republic two countries one one island with similar problems. That did not go over well in some circles because it reflected the reality that the rest of the world sees Haiti's problems as regional rather than isolated by its very porous borders.