Not keen on arguing bias and various media outlets but the bias exhibited by Rasmussen is as glaring as the 'other side' bias by other outlets. Rasmussen often limits their polling to only those who have a landline..that alone produces bias.
BUT, fact remains, albeit w/o specifics, that UBC is favored by a majority of adult Americans..Including a majority of NRA members. As structured, in CO, I just don't see how it's any problem and certainly prevents those who cannot pass the very rudimentary BGC here on Colorado, from buying handgun..BUT, of course, gun shows up in Wyoming are pretty dern close.
Aside from possible bias of the pollster, another problem is that the person being polled may not understand or have a different understanding of what is being asked. On the subject of UBCs, one person may think of it as a simple NICS check, someone else might think it includes a complete psych evaluation, gun registration, etc.
Another frequently asked question is: Do you think we need more gun control laws? If the person being polled is completely ignorant of the current laws, his answer may not be an accurate picture of how many restrictions he thinks are the right balance.
As LeverGunFan pointed out, we really don't know how many NRA members do or do not support UBCs
At any rate, basing public policy on polls is a really bad idea.
On the subject of the UBCs themselves, as others have pointed out, they can form the basis of gun registration which serves no legitimate purpose. They will have zero impact on criminals obtaining them, as they can either buy on the black market or (illegally) have someone do a straw purchase for them. Prohibited persons who currently attempt to buy them from FFLs are almost never prosecuted. What is the point of adding more expense, bureaucracy and potential for abuse to law abiding gun owners with no corresponding benefit for crime reduction?