While we can't know how common weight related humour (aka "Fat Jokes") was in the public, we can take a guess based off how Taft was covered in the media, specifically, news paper political cartoons.
Leading up to the 1908 Election, Taft was favored considerably by the incumbent Theodore Roosevelt. The two had been friends for years and were extremely close. Taft had served as Secretary of War during TR's second term, and would act as the unofficial acting-president when the president was out of town.
Unfortunately, the public wasn't all that too enthused about Taft. Theodore Roosevelt was beloved by many. He was the first president that actively sought media attention. He used his ability to creat, and influence news to achieve his policy goals. Taft, while popular , did not attract the same adoration.
Reinforcing the "poor replacement" belief about Taft, cartoonists depicted the obese Taft as being squeezed into Theodore Roosevelt costumes.
Here, you can see Uncle Sam skeptically looking at Taft dressed as a member of the Rough Riders (the cavalry regiment Roosevelt had founded during the Spanish-American War). Since newspapers were joking about his weight, it's probable that it was a subject the public thought about as well.
Taft is an interesting figure in US political history, as he's probably the only President who didn't really want to be president. Taft was an insanely talented person, but wasn't very strong willed. He served as a lawyer, then Judge in his early career. He excelled as both. His talent attracted a lot of attention by those within the Republican Party, and it wasn't long before he found his way to Washington DC.
But Taft wasn't happy in government. Being a judge was his passion in life, and had hoped to serve on the Supreme Court some day. Due to consistent pressure by those around him, primarily his wife and his Mother, he relented and left his judicial appointment for government.
Unfortunately, Taft went on to be a poor President, especially in the opinion of former President Theodore Roosevelt. Roosevelt was so dissatisfied with Taft's performance as president that he ran as an intentional spoiler candidate in the 1912 election to limit Taft to one term.
Taft would then go on serve as Chief Supreme Court Justice, a job he was both better at, and enjoyed more.
Source: The Bully Pulpit - Doris Kearns Goodwin (2013)
Some more Taft Cartoons:
[Taft and Roosevelt as Small Children] (http://l7.alamy.com/zooms/1f1e48bf6a614380a18ac2178f180e6c/cartoon-showing-president-elect-william-taft-1857-1930-already-sparing-cwa8ke.jpg)
