Howard Zinn writes,
On August 1, 1917, the New York herald reported that in New York City ninety of the first hundred draftees claimed exemption. In Minnesota, headlines in the Minneapolis Journal of August 6 and 7 read: "DRAFT OPPOSITION FAST SPREADING IN STATE" and "CONSCRIPTS GIVE FALSE ADDRESSES."...Senator Thomas Hardwick of Georgia said "there was undoubtedly general and widespread opposition on the part of many thousands...to the enactment of the draft law. Numerous and largely attended mass meetings held in every part of the State proptested against it..." Ultimately, over 330,000 men were classified as draft evaders."
According to this website, the US had a population of 106,021,537 in 1920. This website tells me that there were about 24 million draft cards on record, about 23% of the population in 1918. So let's say about 25% of the population was eligible for the draft in 1920. That would mean about 37,170,738 men could have been drafted. So, around 80% of the population that could be drafted were classified as draft evaders during WWI.
Today, the US population is around 303,256,320 people. Since only men have to register for the draft (and pretending still there's a fifty-fifty split), that means about 151,628,160 people are able to be drafted currently (ignoring age restrictions as I did above). So if the draft were put in place and the US classified the same percentage (6.23%) of the population as draft evaders as it did during WWI, that means that 9,446,434.4 men would be classified as draft evaders. Or, to put it differently, the entire population of New York City (men and women) and then some would be classified as draft evaders.
So imagine a scenario in regards to draft evasion happening today like it did during WWI. War is declared, the draft is invoked, there are enough draft evaders to equal (and go over) the population of New York City. Yet we still go to war.
As for the Vietnam War, Zinn writes,
By mid-1965, 380 prosecutions were begun against men refusing to be inducted; by mid-1968 that figure was up to 3,305. At the end of 1969, there were 33,960 delinquents nationwide.
Zinn goes onto write that anti-war feelings continued to grow in the nation, which eventually (along with other factors) led to US withdraw of troops. But not before the draft was used and not before so many people were killed.
My point is, throughout history there has been resistance to the draft. Politicians know that it is going to be unpopular. But that has not stopped them from using the draft in the past and I think we are being naive if we think that negative public opinion will stop them from using the draft today.