For example, while Republicans are adamant that requiring an ID to vote will prevent the illegal use of an election ballot while also providing a public record of who voted, they are opposed to forcing every gun sale to meet these same standards. Given that a survey of Chicago prisoners showed that the vast majority of criminals obtained their gun from a friend or family member, the objection to making privates sales subject to the rules followed by licensed dealers looks a lot like the NRA is protecting bad guys. Imagine how outraged many of these same people would be if a registered voter were allowed to take a few ballots home for their family and friends and drop them off later without providing any proof that the ballot was filled out by a legal voter.

I don't think republicans as a party or the NRA are opposed to background checks for private party sales. What they are opposed to is putting the burden for those checks on the private parties involved. right now, retail firearms transactions made in the US use The National Instant Criminal Background Check System (NICS). This is a federal system put in place to perform background checks, as the name implies. You need to have a federal firearms license to use this system however. A regular Joe off the street can't call the NICS and request a background check. If the NICS were opened up to private parties, even for a small fee, it would probably see a great deal of use, even if not required. But the federal government probably prefers not to open it up, as it is already strained with the current number of checks it performs. They would also have to put procedures in place to prevent private parties from abusing the system. So if the federal government won't do it, it would fall tot he states. But many states would decline to create their own system (look at how many states reacted to the affordable care act). So while it is easy to say you want background checks for private party transactions, it's not as simple as passing a law requiring it. You need to ramp up the current NICS, or create inter-operable state versions.

Having said all that, even if the NICS is opened up, and background checks are required for all firearms transactions, both retail and private, it will have very little effect on illegal gun acquisitions. Straw purchases are already illegal. This means it is currently a crime for a friend or family to buy a firearm and give it to a prohibited person. In fact, it is also a crime to take money from a person who is not prohibited, and buy a gun for them. More background checks won't change this.

I don't know how to keep guns out of the hands of criminals in the United States. The fact that gun ownership is a protected right means guns are freely available to the average law-abiding adult citizen. Any previously law-abiding citizen can decide to break the law with a gun, or give a gun to a prohibited person. Adding more laws won't change that. You can make the penalties for breaking the law with a firearm harsher (and in some cases they already are), but that won't deter everyone. At some point, you have to accept that a certain amount of firearm crime is always going to exist, and nothing will change that, short of banning guns. A federal gun registry and universal background checks won't change that. Banning certain types of firearms won't change that. If there are guns in the hands of people, there will be crimes committed with guns.