No. Nobody should have to register for the draft.
First of all, even if you still feel that the draft might be necessary, why should the government bother with Selective Service? If you go to public school, the government already has your address and vital data before you even turn 18. If you've never been to public school, you either come from a rich family and they will find a way to keep you stateside or you're Amish and you have contentious objector status. And there are many other ways in this day and age for the government to track you down, unless you are a fugitive or living off the grid. If you are a fugitive, the military wouldn't want you. If you are successfully living off the grid then they should make you a better offer than simply threatening to charge you with draft evasion.
Second of all, would the draft even matter in a age of drones and nuclear weapons? Only suicidal terrorists want to attack the US. Yet foreign terrorists only manage to kill fewer Americans than domestic street gangs. And has anyone bothered to check if these street gang members are registered with Selective Service? Technically, even felons have to register.
Which brings us to another point about the draft: it creates perverse incentives. If healthy people are required to fight, that makes a lot of people want to be unhealthy. Of course the military tests for drugs, so people will tell you to smoke so much dope that your piss turns olive drab. The military has weight limits for recruits, so many young people might put off that diet until their 26th birthday.
Was the draft ever necessary? During the war of 1812, when British troops were setting fire to the White House, America didn't bother with a draft. After the attack on Pearl Harbor and 9/11, recruiting centers were so packed that many men were told to come back later. When America is under attack, American are eager to fight. It's when our government wants to get involved in other countries' squabbles that the military finds itself short of volunteers.